E-book

Content 1
14
Heart of Algebra 18
20
Linear Function 27
Parallel and Perpendicular Lines 29
Linear Function (Table) 31
Linear Word Problems: y = mx + c 32
Linear Word Problems: y = m(x – a) 36
Linear Word Problems: Ax + By = C 42
Linear Word Problems: Using Less than 48
Linear Inequality, System of Linear Inequality Graph 50
Linear Inequality, System of Linear Inequality Word Problems
Solving Linear Equations and System of Linear Equations 54
Linear Inequality 61
Absolute Value 67
74
Passport to Advanced Math 81
85
Operations with Polynomials and Rewriting Expressions 88
Quadratic Equations 95
Quadratic Graph 103
Quadratic Word Problems 110
Exponential Expressions and Radicals 114
Exponential Equation 118
Exponential Graph
Exponential Word Problems
Dividing Polynomials by a Linear Expression
Function Notation
Function Graph
Translation

Problem Solving and Data Analysis 121
129
Ratio, Proportion, Units 135
Percent 141
Interpreting Relationships Presented in Tables 152
Interpreting Relationships Presented in Scatterplots 156
Data Inference 158
Sample and Population 162
Margin of Error 171
Statistics 174
Data Distribution
Boxplot 177
181
Additional Topics 183
185
Circle Equation 190
Complex Numbers
Circle Theorems
Similar Triangles
Speed, Distance, Time

The Ivy 1 Heart of Algebra

Heart of Algebra

Linear Function

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Heart of Algebra 2 The Ivy

Graph In Exercise 2 – 6, sketch the straight line determined by the
points, and find the slope intercept form of the line
1.
What is the slope of the line given by the following 2.
equation? (1, 1), (2, 4)

a) y = 2.5x – 7 y

b) y = 3 x – 12 x
7 7

c) y= 29 x – 2
8 3

d) y = –12.5 + 3.7x

e) y = –20.375 + 101.25x

f) 3x – 4y = 7

g) 2x – 5y = 9 3.
h) 8x – 6y = 1 (1, 2), (3, –2)
i) x + 3y = 10
j) 7x + 2y = 5 y

x

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The Ivy 3 Heart of Algebra
y
4. 6.
(–2, 1), (2, –4) (3, 5), (3, –2)

y

xx

5. 7. March 2019 US
(1, –3), (5, 1) y

y

x 1 x
O2

Which of the following could be an equation for the graph

shown in the xy-plane above?
2
a) y = − 3 x+8

b) y = − 3 x + 4
2
1
c) y = − 3 x + 4

d) y = − 4 x + 8
3

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Heart of Algebra 4 The Ivy

8. June 2016 10.
y
Which of the following represents the graph of the equation
l
4x + 10 = 0?
8
(6, a) a) y

O 3x 6824 24 68 x
In the xy-plane above, what is the value of a? –––

a) 6 –––
b) 4.5
c) 4 b) y
d) 3
6824 x
––– 24 68

–––

9. c) y

y 8624 24 68 x
–––
5 x
4 –––
3
–––– 2 d) y
1
6842 x
12345 –––
–
– –––
–
–

Which of the following equations represents the line 24 68
graphed above in the xy-plane?

a) x = –4
b) y = –4
c) y = x – 4
d) x = y – 4

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The Ivy 5 Heart of Algebra

11. 13. 2
3
y 4y = x–8

5 Which of the following is the graph of the equation above in
4
3 the xy-plane?
2
1x a) y

–––– – 12345 8624
–––
–
– ––– x
– 24 68

A line is graphed in the xy-plane shown above. b) y
Which of the following is an equation of the line?
6842 24 68 x
a) 3.5y = 0 –––
b) 3.5x = 0
c) y – 3.5 = 0 –––
d) x – 3.5 = 0

12. c) y

y 6824
–––
7 24 68 x
6 –––
5 x
4
3
2
1

–– –– – – 1234567 d) y

– 8642 24 68 x
– –––
–
– –––
–
–

A line is graphed in the xy-plane shown above.

Which of the following is an equation of the line?
3
a) y = 2 x

b) y = 2 x
3
3
c) y = − 2 x

d) y = − 2 x
3

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Heart of Algebra 6 The Ivy

14. March 2019 US 15.
ax + by = b
y
In the equation above, a and b are constants and 0 < a < b.
Which of the following could represent the graph of the 6 x
equation in the xy-plane? 5
4
a) y 3
2
1

–– –––– – 1234567
–
1 x –
1 –
–

Which of the following equations represents the line

graphed above in the xy-plane?
3
b) y a) y = 7 x+1

b) y = 3 x–1
7
1 3
1 x c) y = − 7 x + 1

d) x = − 3 y – 1
7

c) y 16. December 2018
y
1 x I
1 II x

O IV
III

d) y The graph (not shown) of a linear function, h, where
y = h(x), is a line completely contained in only quadrant I
1 x and II. Which of the following could define the function h?
1
a) h(x) = 3x + 3

b) h(x) = 3x

c) h(x) = 3

d) h(x) = –3

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The Ivy 7 Heart of Algebra

17. 19.

y y

7 –––– –––– 54321 1 2 3 4 5 6 7 8 9 x
6
5 x
4 1234567
3
2 ––––
1
–––– – – Which of the following equations represents the line
–––
–
–
–

A line is graphed in the xy-plane shown above. graphed above in the xy-plane?

Which of the following is an equation of the line? 4
3
a) x + y – 1 = 0 a) y + x = 7

b) x + y + 1 = 0 b) y – 4 x = 7
3
c) x – y + 1 = 0 4
3
d) x – y – 1 = 0 c) y + x = –7

d) y – 4 x = –7
3

18. 20.

y y

5 x 7654321 12345678 x
4 ––– –– – –
–––– 3 12345
2 –––––––
1
–
–
–
–

A line is graphed in the xy-plane as shown above. A line is graphed in the xy-plane above.
Which of the following is an equation of the line?
Which of the following is an equation of the line?
a) y = –4x – 2 3
b) y = 4x – 2 a) y = 8 x + 2
c) x = –4y – 2
d) x = 4y – 2 b) y = 3 x – 2
8
3
c) y = − 8 x + 2

d) y = − 3 x – 2
8

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Heart of Algebra 8 The Ivy

21. March 2018 23.
y
Which of the following graphs represents the equation
3
t (y + 6) = − 2 (x – 8)?
(x, 2)
a) y

Ox 8
6
42
In the xy-plans above, line t passes through the origin and 2468 x
1
has a slope of 2 . What is the value of x? ––– –

––

b) y

8
6
42
2468 x
––– –

––

22. October 2019 US c) y
y
8
k 6
42
Ox 2468 x
In the xy-plane above, line k passes through the points (0, 2) ––– –
and (3, 0). If line k is defined by the equation y = mx + b,
where m and b are constants, what is the value of b? ––

d) y

8
6
4
2
2468 x
––– –

––

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The Ivy 9 Heart of Algebra

24. 25.
y + 3 = 2(x + y) + 8 x= y+
Which of the following is the graph of the equation above in Which of the following could be the graph of the equation
the xy-plane? above?

a) y a) y
864 864

–8–6–4 2 4 6 8 x –8–6–4 2 4 6 8 x
–––468 –––468

b) y b) y
864 864

–8–6–4 2 4 6 8 x –8–6–4 2 4 6 8 x
–––468 –––468

c) y c) y
864 864

–8–6–4 2 4 6 8 x –8–6–4 2 4 6 8 x
–––468 –––468

d) y d) y
864 864

–8–6–4 2 4 6 8 x –8–6–4 2 4 6 8 x
–––468 –––468

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Heart of Algebra 10 The Ivy

26. 27.
y + = 3(–x + y + 2) Which of the following equations represents a line in the xy-
Which of the following could be the graph of the equation plane with an x-intercept at (–2, 0) and a slope of 4?
above?
a) y = 4x + 8
a) y b) y = –4x + 8
864 c) y = 4x – 2
d) y = –4x – 2
–8–6–4 2 4 6 8 x
–––468 28. 1
5
b) y The equation y–2= (x + 5) is graphed in the xy-plane.
864
Which of the statements below is true of its graph?
–8–6–4 2 4 6 8 x
–––468 a) The graph has a slope of –5 and a y-intercept of 5.

c) y b) The graph has a slope of –5 and passes through the
864
point (2, –5). 1
–8–6–4 2 4 6 8 x 5
–––468 c) The graph has a slope of and passes through

d) y the point (–5, 2). 1
864 5
d) The graph has a slope of and y-intercept of 5.
–8–6–4 2 4 6 8 x
–––468

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The Ivy 11 Heart of Algebra

29. 31. May 2018

The equation 6 – 2y = x is graphed in the xy-plane. In the xy-plane, line L passes through the point (0, 1) and

Which of the following is a true statement about the graph? (1, 4). Which of the following is an equation of line L?
1
a) The graph’s x-intercept is 6 and its y-intercept is –2. a) y = 3 x + 1

b) The graph’s x-intercept is 6 and its y-intercept is 2. b) y = 1 x – 1
3
c) The graph has a slope of –2. c) y = 3x + 1
1
d) The graph has a slope of − 2 . d) y = 3x – 1

30. 32. March 2019
In the xy-plane, line k has a slope of –3 and passes through
The equation 2x – y = 8 is graphed in the xy-plane. the point (–2, 7). Which of the following is an equation of
line k?
Which of the following is a true statement about the graph?
a) 3x + y = 1
a) The graph’s x-intercept is 4 and its y-intercept is –8. b) 2x + y = 3
c) –2x + y = 11
b) The graph’s x-intercept is 4 and its y-intercept is 8. d) –3x + y = 13

c) The graph has a slope of –2.
1
d) The graph has a slope of 2 .

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Heart of Algebra 12 The Ivy

33. May 2018 35.

In the xy-plane, the graph of line L has a slope of 3. Which of the following is an equation of the line in the
5 10
Line K is parallel to line L and contains the point (3, 10). xy- plane with x-intercept 3 and y-intercept 7 ?

Which of the following is an equation of line K? a) –7y + 6x = 10
1
a) y = − 3 x + 11 b) 7y + 6x = –10

b) y = 1 x + 9 c) 7y + 6x = 10
3
c) y = 3x + 7 d) 7y – 6x = 10

d) y = 3x + 1

34. 36. March 2019 US
In the xy-plane, line k intersects the y-axis at the point (0, –6)
Which of the following is an equation of the line in the and passes through the point (2, 2). If the point (20, w) lies
2 8 on line k, what is the value of w?
xy- plane with x-intercept − 7 and y-intercept 7 ?

a) 7y – 28x = 8

b) 7y + 28x = 8

c) –7y + 28x = 8

d) –7y – 28x = 8

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The Ivy 13 Heart of Algebra

37. The line graphed at left in the xy-plane can be written as the equation
y = mx + b, where m and b are nonzero constants. Which of the
y following represents the graph of x = my + b?

8

6

4

2 2 4 6 8x
–––

–

–

–

–

a) y b) y

8642 24 68 x 8642 24 68 x
–8 –6 –4 –8 –6 –4

–––468 –––468

c) y d) y

8642 24 68 x 8642 24 68 x
–8 –6 –4 –8 –6 –4

–––468 –––468

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Heart of Algebra 14 The Ivy

Parallel and Perpendicular Lines

1. 3 2.
2
The equation y = (x – 8) is graphed in the xy-plane. Which of the following is an equation of the line in the xy-

Which of the following equations will have a graph that is plane that passes through the point (2, 6) and is parallel to
2
parallel to the graph of the above equation and have an the line with equation x = 3 y + 2?

x-intercept on the negative x-axis? a) y = − 2 x + 3
3 3
a) y = 2 (x + 8) 2
b) y = − 3 x – 3
3
b) y = 2 x – 8 c) y= 3 x+3
2
c) y = − 2 (x + 8) 3
3 d) y= 2 x–3
2
d) y = − 3 x – 8

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The Ivy 15 Heart of Algebra

3. 5. May 2017 US

Which of the following is an equation of the line in the y

xy-plane that passes through the point (1, 1) and is parallel to 8

the line with equation y = –3x – 1? 6

1 4
a) y = 3 x + 4
1 2
3
b) y = x + 4 – –– – 2 4 6 8x

c) y = –3x + 4 –

d) y = –3x + 1 –

–

–

A line is shown in the xy-plane above. A second line (not

shown) is parallel to the line shown and passes through the

points (1, 1) and (3, c), where c is a constant. What is the

value of c?

4. 6.

The equation 2y + 3x = –10 is graphed in the xy-plane. The equation y = x + a is graphed in the xy-plane, where a is
a positive constant. A second equation, y = x – a is graphed
Which of the following equations has a graph that is parallel in the same xy-plane. How do the slope and y-intercept of
the graph of the first equation compare with the slope and
to the graph of the above equation? y-intercept of the graph of the second equation?
2 5
a) y = − 3 x + 2 a) The slopes are the same; the y-intercept of the first
equation’s graph is the point (0, a), and the
b) y= 2 x –5 y-intercept of the second equation’s graph is the
3 point (0, –a).
3
c) y= 2 x –5 b) The slopes are the same; the y-intercept of the first
equation’s graph is the point (0, a), and the
d) y = − 3 x + 5 y-intercept of the second equation’s graph is the
2 2 point (0, a).

c) The y-intercepts are the same; the slope of the first
equation’s graph is a, and the slope of the second
equation’s graph is –a.

d) The y-intercepts are the same; the slope of the first
equation’s graph is –a, and the slope of the second
equation’s graph is a.

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Heart of Algebra 16 The Ivy

7. 9.

The graph of y = mx + b in the xy-plane is a line, and the The equation 6y + 12x = 18 is graphed in the xy-plane.
graph of y = m(x + 1) + b – 2 in the xy-plane is also a line.
How do the slope and y-intercept of the first line compare Which of the following equations has a graph that is
with the slope and y-intercept of the second line?
perpendicular to the graph of the above equation?
a) The slopes and the y-intercepts are the same for all
values of m. a) y = –2x + 3
1
b) The slopes are the same; the y-intercept of the first b) y = 2 x + 3
line is at the point (0, b),
and the y-intercept of the second line is at the point c) y = 2x + 3
(0, b – 2). 1
d) y = − 2 x + 3
c) The slopes are the same; the y-intercept of the first
line is at the point (0, b),
and the y-intercept of the second line is at the point
(0, m + b – 2).

d) The y-intercepts are the same; the slope of the first
line is 1 less than the slope of the second line.

8. 10.

Which of the following is an equation of the line A graphed A line in the xy-plane passes through the point (4, –1) and is
in the xy-plane that passes through the point (–1, 3.5) and is perpendicular to the line with equation y = x + 5. Which of
perpendicular to the line B whose equation is x + 4.5 = 0? the following is an equation of the line?

a) x = –1 a) y = x + 3
b) x = 3.5 b) y = x – 3
c) y = 3.5 c) y = –x + 3
d) y = 4.5 d) y = –x – 3

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The Ivy 17 Heart of Algebra

11. 13. December 2017
y
Which of the following is an equation of the line in the

xy-plane that passes through the point (–5, 3) and is
1
perpendicular to the line with equation y = – 8 x + 6?

a) y = – 1 x + 43
8
b) y = 8x + 43 1
1 O1 x
c) y = – 8 x – 43

d) y = 8x – 43

The graph of line k is shown in the xy-plane to the left.

Which of the following is an equation of a line that is

perpendicular to line k?

a) y = –2x + 1
1
b) y = − 2 x + 2

c) y = 1 x + 3
2
d) y = 2x + 4

12.

Which of the following is an equation of the line in the xy-

plane that passes through the point (3, –4) and is 14. October 2017 US
1 y
perpendicular to the line with equation y = 4 x – 7?

a) y = 1 x + 8
4
1
b) y = 4 x – 8 3
2 y = f (x)
c) y = –4x + 8

d) y = –4x – 8

1

– –O 1 2 3x

The graph of the linear function f is shown in the xy-plane

above. The graph of the linear function g (not shown) is

perpendicular to the graph of f and passes through the point

(1, 3). What is the value of g(0)?

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Heart of Algebra 18 The Ivy

Linear Function (Table)

1. October 2018 US 3. March 2019

x a 3a 5a xy
74
y 0 –a –2a 4 10
1 16
Some values of x and their corresponding values of y are The table above shows the coordinates of three points on a
shown in the table above, where a is a constant. If there is a line in the xy-plane. What is the value of x when y = 0?
linear relationship between x and y, which of the following
equations represents the relationship?

a) x + 2y = a
b) x + 2y = 5a
c) 2x – y = –5a
d) 2x – y = 7a

2. October 2018 4. May 2018

x h(x) x f (x)
38 8 12
5 12 12 17
The table above shows two values of h(x) for the linear The table above shows two pairs of values for the linear
function h. Which of the following equations defines h? function f. The function can be written in the form
a) h(x) = 0.5x – 1 f (x) = ax + b, where a and b are constants.
b) h(x) = 2x + 2 What is the value of a + b?
c) h(x) = 5x + 12
d) h(x) = 3x + 8

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The Ivy 19 Heart of Algebra

5. March 2019 6. December 2017

xy Pages Total paid
9 $308
02 12 $341
15 $374
30 18 $407

The table above shows the coordinates of two points that lie The table above shows the total amount of money m, in
dollars, a comic book artist is paid for completing a project
on line l in the xy-plane. If the slope of line l is m, what is consisting of p pages. The formula m = rp + k shows the
relationship between the variables m and p, where r is the
the value of m? amount of money, in dollars, paid per page and k is the one-
3 time fee the artist is paid for the project.
a) − 2 What is the value of r?

b) − 2
3
2
c) 3

d) 3
2

Function

1. October 2019 US 2. October 2018
For the linear function t, t(0) = 8 and t(4) = –4.
The function f is linear, f (2) = 17, and f (8) = 19. What is the value of t(1)?

If f (x) = mx + b, where m and b are constants, what is the a) –3
b) 2
value of b? c) 3
d) 5
a) 11

b) 13
49
c) 3

d) 55
3

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Heart of Algebra 20 The Ivy

Linear Word problems: y = mx + c form

1. October 2018 3.

The distance d, in feet, from the ground to a flag on a E = –2.4t + 75
flagpole t seconds after a person begins lowering it is
modeled by the function d(t) = 20 – 2t. In this function, A big cruise ship drops anchor off the Caribbean island of
what does 20 represent? Antigua. The heavy anchor drops into the water at a constant
rate. The equation shown above gives the anchor's elevation,
a) The distance, in feet, of the flag from the ground E, measured in meters relative to the water's surface, after t
before it is lowered seconds. How many meters does the anchor drop every 5
seconds?
b) The distance, in feet, of the flag from the ground
after 2 seconds

c) The number of seconds it takes for the flag to
reach the ground

d) The rate, in feet per second, at which the flag
moves as it is lowered

2. 4.

D = –200t + 9000 S = –0.25t + 4
Harry took a loan from the bank. Each month, he pays a fixed A lake near the Arctic Circle is covered by a thick sheet of
amount of money back to the bank. The equation above shows ice during the cold winter months. When spring arrives, the
the remaining amount of the loan, D, measured in dollars, warm air gradually melts the ice, causing its thickness to
after t months. How much does Harry pay back to the bank decrease at a constant rate. The equation shown above gives
each month, in dollars? the ice sheet's thickness, S, measured in meters, t weeks after
it begins to melt. By how many meters does the sheet's
thickness decrease every 6 weeks?

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The Ivy 21 Heart of Algebra

5. 7. l
2
T = 29 – 8a w= 3.1 +
The temperature, T, in degrees Celsius, as measured by a
weather balloon at an altitude of a kilometers from the The weight, w, in ounces, of a glass soda bottle design with
ground is given by the equation above. What is the best
interpretation of 29 as shown in the equation above? a side length of l inches is given by the equation above.

a) The temperature on the ground is 29 degrees By how many ounces does the weight of the bottle increase
Celsius.
for every inch of increase in the side length?
b) The weather ballon reaches a maximum altitude of
29 kilometers. a) 0.5

c) The temperature increases by 29 degrees Celsius b) 2
for every kilometer increase in altitude.
c) 3.1
d) The temperature decreases by 29 degrees Celsius
for every kilometer increase in altitude. d) 3.6

6. 8.

D = –38t + 220 D = 1,875 – 0.55t
Rachel is driving a race car at a constant speed on a closed The distance, D, in meters, between an antarctic glacier and
course. The formula shown above describes the remaining the coast t days after January 1, 2010 is approximated by the
distance, D, measured in meters, that Rachel has to travel after equation above. How does the distance between the glacier
t seconds. What is Rachel's speed in meters per second? and the coast change over time?

a) The glacier moves 0.55 meters per day closer to
the shore.

b) The glacier moves 1,874 meters per day closer to
the shore.

c) The glacier moves 0.55 meters per day further
from the shore.

d) The glacier moves 1,874 meters per day further
from the shore.

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Heart of Algebra 22 The Ivy

9. 11. May 2018

H = 1.88L + 32.01 The front row of an auditorium has 10 seats. There are 50
The formula above can be used to approximate the height H, rows in total. If each row has 2 more seats than the row
in inches, of an adult male based on the length L, in inches, before it, which expression gives the total number of seats in
of his femur. What is the meaning of 1.88 in this context? the last row?

a) The approximate femur length, in inches, for a a) 10 + 2(50 – 1)
man with a height of 32.01 inches b) 10 + 2(50)
c) 50(10 + 2)
b) The approximate increase in a man’s femur length, d) 10 + 2(50)2
in inches, for each increase of 32.01 inches in his
height

c) The approximate increase in a man’s femur length,
in inches, for each one-inch increase in his height

d) The approximate increase in a man’s height, in
inches, for each one-inch increase in his femur
length

10. May 2018 US 12. October 2018
Larry is training for a half marathon by going on a long run
The equation p = 14.7 + 0.439d approximates the pressure p, every Saturday. He will run 3 miles on the first Saturday of
his training. Each Saturday after the first, he will run 2 more
in pounds per square inch, exerted on a diver at a depth of d miles than he ran on the preceding Saturday. Which of the
following equations represents the number of miles m Larry
feet (ft) below the surface of the water. What is the increase will run on the nth Saturday of his training?

in depth that is necessary to increase the pressure by 1 pound a) m = 2n + 3

per square inch? b) m = 2n + 1
1
a) 0.439 ft c) m = 2n – 1

b) 1 ft d) m = 2n + 3
14.7
c) 0.439 ft

d) 14.7 ft

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The Ivy 23 Heart of Algebra

13. 15. d
25
E = 420 – 7s l = 10 +
The elevation, E, in meters, of a skydiver s seconds after
parachute activation is given by the equation above. What is In a study, the nails of 89 guinea pigs were trimmed to the
the best interpretation of 420 as shown in the equation?
same length and measured over the course of two months.
a) The skydiver started diving at an elevation of 420
meters. The average nail length, l, in millimeters, d days after the

b) The skydiver activated the parachute at an start of the study is approximated by the equation above.
elevation of 420 meters.
What is the best interpretation of 25 as shown in the
c) The skydiver took 420 seconds after parachute
activation to reach the ground. equation above?

d) The skydiver’s elevation decreased by 420 meters a) The average nail length started at 25 millimeters.
per second after parachute activation.
b) The average nail length increased by 25

millimeters each day.

c) The average nail length reached a maximum of 25

millimeters.

d) The average nail length increased by 1 millimeter

every 25 days.

14. October 2019 16. 1
4
d(x) = –30x + 87 d = – t + 8.5
The function d above models a soccer ball’s distance, in feet,
from the goalie x seconds after a player passed the ball, The average depth of ice, d, in inches, on a frozen lake at a
where the ball remains on the ground and 0 ≤ x ≤ 15.
Which of the following is the best interpretation of the time of t days after March 1, 2011 is approximated by the
y-intercept of the graph in the xy-plane of y = d(x)?
equation above. What is the best interpretation of 4 as shown
a) The soccer ball was passed at a distance of
approximately 30 feet from the goalie. in the equation above?

b) The soccer ball was passed at a distance of a) The depth of ice on March 1, 2011 is 4 inches.
approximately 87 feet from the goalie.
b) The depth of ice reaches 0 inches after 4 days.
c) The soccer ball was moving toward the goalie at a
speed of approximately 30 feet per second. c) The depth of ice decreases by 4 inches every day.

d) The soccer ball was moving toward the goalie at a d) The depth of ice decreases by 1 inch every 4 days.
speed of approximately 87 feet per second.

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Heart of Algebra 24 The Ivy

17. d 19. 2
3 5
V = –2.6 + P = m + 1.5

The electric potential, V, in volts, between two metal plates a The price, P, in dollars, of a train ticket used to travel a

distance of d millimeters from the left plate is given by the distance of m miles is given by the equation above. By how

equation above when 0 ≤ d ≤ 15. By how many millimeters many miles does the traveling distance increase if the ticket

does the distance from the left plate increase for the potential price increases by 1 dollar?

to increase by 1 volt?

18. 20.

h = 11.2 – 0.125d A construction company's fees include a flat daily charge to
come to a work site, in addition to a constant hourly rate. Any
The ceiling height, h, in feet, for a particular room in a house fractional hours are charged at that fraction of the hourly rate.
a distance of d feet from the west wall is given by the equation Additionally, if a job takes less than 10 hours to complete, it
above. In order for the ceiling height to decrease by 1 foot, is completed in one day. According to these rules, a 3.5-hour
how much further from the west wall does the distance change job would cost $250 and an 8-hour job would cost $475. If
in feet? the construction company completes a job in one day, which
of the following functions best models the cost, c, in dollars,
a) 0.125 for an h hour job?
b) 8
c) 11.2 a) c(h) = 50h
d) 89.6 b) c(h) = 60h
c) c(h) = 50h + 75
d) c(h) = 60h + 10

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The Ivy 25 Heart of Algebra

21. 23.
C = 1,500n + 1,000
A certain meteorological table shows that when the The cost, C, in dollars, to ship nnnn cars from Europe to the
temperature is 59 degrees Fahrenheit (°F) at sea level, the United States is given in the equation above. What is the
temperature is 48.3 °F at 3000 feet (ft). If air temperature is difference, in cost, to ship 7 cars versus 6 cars?
a linear function of altitude, which of the following best a) $500
models the air temperature, T, in °F, at an altitude of h feet, b) $1,000
for a day when the temperature is 59 °F at sea level? c) $1,500
d) $2,500
a) T(h) = 59 – 3.6h
b) T(h) = 59 – 10.7h
c) T(h) = 59 – 0.0107h
d) T(h) = 59 – 0.0036h

22. 24.

Anthropologists have noticed that they can predict the height T = 277 + 4s
of a human being based on the length of their femur. The The temperature, T, in degrees Kelvin, of a resistor in a circuit
relationship between the person's height, H, in centimeters, s seconds after connecting a battery is given by the equation
and length of the femur, f, in centimeters, can be modeled by above. How many seconds does the battery remain connected
the equation below. for the temperature to increase by one-degree Kelvin?

H = 2.72f + 42.12 a) 0.25
One anthropologist found the femurs of two different b) 4
people. One femur measured 22 centimeters and the other c) 69
femur measured 23 centimeters. What is the difference in d) 277
their heights?

a) 1 centimeter
b) 2.72 centimeters
c) 27.2 centimeters
d) 42.12 centimeters

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Heart of Algebra 26 The Ivy

25. 26. December 2018
w = 500 – 6.2t D
A rain barrel is being used during the dry season to water a
garden with previously collected rainwater. With a hose 2.5
connected to its outlet, the above equation approximates Distance from home
amount of rainwater in the barrel, w, in liters, after t minutes (miles) 00 30 t
of watering. What is the meaning of 6.2t in this equation?
a) It takes 6.2t minutes to empty the barrel. Time (minutes)
b) 6.2t minutes are needed to fill the barrel. Miranda jogged home from a friend’s house along a straight
c) 6.2t liters of rainwater is emptied after t minutes.
d) There are 6.2t liters in the barrel when filled for t path without making any stops. The graph above gives

minutes. Miranda’s distance D, in miles, from home as a function of

the time t, in minutes, after she left her friend’s house.

Which of the following statements is true about Miranda’s

jog home?

a) Miranda’s speed, in miles per minute, decreased at

a constant rate.

b) After jogging for approximately 2.5 miles,

Miranda was 30 minutes from home.

c) Miranda’s distance from home decreased at a

constant rate of 12 miles per minute.

d) Miranda’s distance from home decreased at a

constant rate of 1 mile per 12 minutes.

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The Ivy 27 Heart of Algebra

Linear Word problems: y = m(x – a) form 3.
P = 47.4(G – 174.5)
1. The profit, P, in dollars, to an amusement park serving G
I = 0.5(18 + m) guests over one day is given by the equation above. What is
A landlord determines that the total amount of income, I, in the minimum number of guests that need to be served in
thousands of dollars generated by renting out a home is order to make a positive profit?
given by the equation above. Negative values of m represent a) 48
the time m months before January 1, 2010 and positive b) 174
values of m represent the time m months after January 1, c) 175
2010. The landlord had an income of 0 when the home was d) 8,272
bought and only began receiving income from rent
afterward. When did the landlord buy the home? 4.
a) The landlord bought the house 18 months before d = 1.06(212 – t)
A spacecraft heading for Pluto will take pictures of several
January 1, 2010. other planets on its way. The above equation gives the
b) The landlord bought the house 9 months before distance, d, in millions of kilometers, of the craft from its
first photo opportunity with Mars in terms of the time, t, in
January 1, 2010. days. What is the meaning of the 212 in this equation?
c) The landlord bought the house 9 months after a) The spacecraft travels 212 kilometers per day.
b) It will take the spacecraft 212 days to reach the
January 1, 2010.
d) The landlord bought the house 18 months after Mars photo opportunity.
c) The spacecraft begins its journey 212 kilometers
January 1, 2010.
from the Mars photo opportunity.
2. d) After 212 days, the spacecraft will be within 1.06
C = 0.25(L – 1200)
The cost, C, in dollars, to the holder of a liability insurance kilometers of the Mars photo opportunity.
account given a total liability of L dollars is given by the
equation above as long as C is positive. By how many
dollars does the liability increase when the cost to the holder
increases by 1 dollar?

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Heart of Algebra 28 The Ivy

5. 7.
M = 8(4.5 – C)
n = 38(25 – d) The magnetic field, M, in microtesla, at the center of a loop
The number of meat patties, n, remaining in a restaurant with a current of C milliamps clockwise is given by the
freezer d days after restocking the freezer is approximated equation above. By how many microteslas does the
by the equation above. What is the best interpretation of 38 magnetic field change for a 1 milliamp increase in the
as shown in the equation above? current?
a) –8
a) The restaurant had 38 meat patties immediately b) –4.5
after restocking. c) 28
d) 36
b) The restaurant had no more meat patties 38 days
after restocking.

c) The number of meat patties increases by 38 each
day after restocking.

d) The number of meat patties decreases by 38 each
day after restocking.

6. 8.

v = 14.2(12.5 – s) B = 11.5(132 – n)
The velocity, v, in meters per second heading west, of an The monthly electricity bill, B, in dollars, for a museum
airplane s seconds after landing is given by the equation using n solar panels is given by the equation above. What is
above. What is the best interpretation of 14.2 as shown in the the best interpretation of 132 in the equation above?
equation above?
a) The museum is currently using 132 solar panels.
a) The plane takes 14.2 seconds to stop. b) The monthly electric bill is 132 dollars without
b) The position of the plane changes 14.2 meters
solar panels.
every second. c) The electric bill is 0 dollars when the museum uses
c) The velocity changes by 14.2 meters per second
132 solar panels.
every second. d) The electric bill decreases by 132 dollars for each
d) The velocity of the plane just as it lands is 14.2
additional solar panel.
meters per second.

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The Ivy 29 Heart of Algebra

9.

m = 0.15(8.6 – s)
The amount, m, in moles, of a chemical s seconds after adding
it to a reactant is given by the equation above. What is the best
interpretation of 8.6 as shown in the equation above?

a) The initial amount of chemical was 8.6 moles.
b) The amount of chemical decreased by 8.6 moles

every second.
c) The chemical is gone 8.6 seconds after adding it to

reactant.
d) The chemical starts changing 8.6 seconds after

adding it to the reactant.

Linear Word problems: Ax + By = C form 2.

1. March 2019 36a + 28.3w = 100
Oona purchased p pounds of peanuts and a pounds of A bottle contains 100 milliliters of a solution of alcohol and
almonds. The peanuts cost $2.45 per pound, and the almonds water at room temperature. The equation above shows the
cost $3.15 per pound. Oona spent a total of $14.35 on possible amounts of alcohol, a, and water, w, in ounces.
peanuts and almonds. The equation below represents this What is the volume of 1 ounce of alcohol at room
situation. temperature in milliliters?

2.45p + 3.15a = 14.35 a) 0.03
What is the meaning of the term 3.15a in this context? b) 2.8
c) 28.3
a) The total cost of almonds in Oona's purchase d) 36
b) The total cost of peanuts in Oona's purchase
c) The total pounds of almonds Oona purchased
d) The total pounds of peanuts Oona purchased

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Heart of Algebra 30 The Ivy

3. October 2019 5. May 2019
H83anonfahhissasvaleasry25eaocfhhmerosnatlha.rTyoegaecthhemr,otnhtehy, asnadveWaytaotttaslaovfes
$1,200 each month. If h and w represent Hannah’s and A petting zoo sells two types of tickets. The standard ticket,
Wyatt’s monthly salaries, in dollars, respectively, which of for admission only, costs $5. The premium ticket, which
the following describes a relationship between monthly includes admission and food to give to the animals, cost $ 12.
salaries? One Saturday, the petting zoo sold a total of 250 tickets and
collected a total of $2,300 from ticket sales. Which of the
a) h + w = 1,200 following systems of equations can be used to find the number
b) 2h + 3w = 1,200 of standard tickets, s, and premium tickets, p, sold on that
c) 15h + 16w = 48,000 Saturday?
d) 16h + 15w = 48,000
a) s + p = 250
\ 5s + 12p = 2,300

b) s + p = 250
12s + 5p = 2,300

c) 5s + 12p = 250
s + p = 2,300

d) 12s + 5p = 250
s + p = 2,300

4. October 2019 US 6. October 2018 US

The City Transit bus line charges $2 for an adult and $1 for Lynne has $8.00 to spend on apples and oranges. Apples
a child to ride one way. During a certain 4-hour shift, a bus cost $0.65 each, and oranges cost $0.75 each. If there is no
driver collects $1,171 from 617 riders. Which of the tax on this purchase and she buys 5 apples, what is the
following systems of equations could be used to determine maximum number of whole oranges she can buy?
the number of adult riders, A, and the number of child riders,
C, during this 4-hour shift?

a) 2A + C = 4(1,171)
A + C = 4(617)

b) 4(2A) + 4(C) = 1,171
4(A + C) = 617

c) 2A + C = 617
A + C = 1,171

d) 2A + C = 1,171
A + C = 617

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The Ivy 31 Heart of Algebra

Linear Word problems: Using Less than

1. 3.
Fabio and Carlos play on a basketball team together. In the
On a 40 °F day, a 25 mile per hour (mph) wind creates a last game, Fabio had 7 points less than 2 times as many points
as Carlos. Fabio scored 31 points in the game. How many
wind chill of 29 °F. To approximate the wind chill at various points did Carlos score in the last game?

temperatures caused by a 25-mph wind, Elon uses the 4. May 2017 US
4 For 5 consecutive even integers, the sum of the first and
following rule: take 24 less than 3 of the air temperature third integer is 20 less than 3 times the fourth integer.
What is the fifth integer?
(°F). If a 25-mph wind creates a wind chill of –5 °F, which
a) 12
of the following best approximates the corresponding air b) 14
c) 16
temperature? d) 26

a) –30.7 °F

b) 14.3 °F

c) 21.7 °F

d) 25.3 °F

2. May 2019

The acceleration due to gravity, in meters per second per
second (m/s2), on Earth is 1.3 m/s2 less than 3 times the
acceleration due to gravity on Mercury. If the acceleration
due to gravity on Earth is 9.8 m/s2, what is the acceleration
due to gravity, in m/s2, on Mercury?

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Heart of Algebra 32 The Ivy

Linear Inequality, Systems of Linear Inequalities Graph

1. March 2019 2. May 2018
0≤x≤y≤6
x ≥ –2x + 11
What is the area, in the xy-plane, of the region consisting of x > 3x – 9
all points that satisfy the inequality above? In the xy-plane, point A is contained in the graph of the
solution set of the system of inequalities above. Which of
the following could be the coordinate of point A?
a) (2, 1)
b) (3, 1)
c) (4, 5)
d) (6, 6)

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The Ivy 33 Heart of Algebra

3. 5.
4x – y < 2 3x – 2y < 3
3x + y < –1 –3x + 2y < –8
Graph the system of inequalities above?
Graph the system of inequalities above?

4. 6. December 2018
x – 2y < 5
–x + 2y > 2 y
10
Graph the system of inequalities above? 8
64
2
– – – – – – 2 4 6 8 10 x
–
–
–
–

The shaded region shown above represents the solution to
which of the following systems of inequalities?

a) y < x
y≤4

b) y < x
y≥4

c) y > x
y≤4

d) y > x
y≥4

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Heart of Algebra 34 The Ivy

7. December 2017 8. October 2017 US
y≥x+2
5y + 3x ≤ 15 2x + 3y ≤ 6

The solution set to the inequality above is represented by the In which of the following does the shaded region represent
line and shaded region of the graph in which of the the solution set in the xy-plane to the system of inequalities
following xy-planes? above?

a) y a) y

4 4
3
2 x –O 4x
1

–O 1 2 3 4 5

b) y –

4 b) y
3 4
2
1 x – O 4x
c) 4x
–O 1 2 3 4 5 – 4x
y
c) y 4

4 – O
3 d)
2 –
1 y
–5 –4 –3 –2 –1 O 1 x 4

d) y –O

4
3
2
1
–5 –4 –3 –2 –1 O 1 x

–

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The Ivy 35 Heart of Algebra

Questions 9 and 10 refer to the following information. 10. October 2018

20 y Music File and E-book Purchases If Kris purchased 6 music files and a number of
e-books, which of the following is NOT a number of
18 e-books she could have purchased?
16
Number of e-books14 a) 2
1102 II I b) 4
Pressure (GPa) c) 8
8 IV l d) 9
6 68
42 III V x
00 24 10 12 14 16 18 20 22 24 26
Number of music files
A website sells only music files and e-books. The website

sells music files for $1 each and e-books for $2 each. Kris

purchases fewer than 18 music files and e-books combined.

The total purchase price is greater than $10 but less than Question 11 refer to the following information.

$25. The dashed lines in the xy-plane above represent these 1.0 AlumPihnaosseilDiciaatgeraPmolyfomrorphs

three constraints on her possible purchases from the site,

where x represents the number of music files and y 0.8 kyanite (795, 1.0)

represents the number of e-books.

0.6 sillimanite
0.4 (500, 0.38)

0.2 andalusite (760, 0)
(350, 0.19) 700 800

0 400 500 600
Temperature ( C)

11. May 2018

Which of the following systems of inequalities best

describes the region where sillimanite can form?

a) P0.0021T – 0.67

P0.0013T – 0.25

9. October 2018 b) P0.0021T – 0.67

Which region of the graph contains all possible P–0.0015T + 1.13
combinations for the number of music files and e-books Kris
could purchase given the constraints? c) P0.0013T – 0.254

a) II P–0.0015T + 1.13
b) III
c) IV d) P0.0013T – 0.25
d) V
P0.0015T + 1.13

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Heart of Algebra 36 The Ivy

12. December 2016
In the xy-plane, which of the following does NOT contain
any points that are part of the solution set to 3x – 4y > 12?

a) The region where x < 0 and y > 0
b) The region where x < 0 and y < 0
c) The region where x > 0 and y > 0
d) The x-axis

Linear Inequality, Systems of Linear Inequalities Word Problems

1. March 2019 2. October 2019 US
In ancient Egypt, from 2810 BC to 2800 BC. the course of
the Nile riverbed moved eastward at least 2 meters per year A certain elephant weighs 200 pounds at birth and gains
but no more than 3 meters per year. Which of the following more than 2 but less than 3 pounds per day during its first
inequalities represents all possible values for the total year. Which of the following inequalities represents all
distance d, in meters, the Nile riverbed moved eastward for 4 possible weights w, in pounds, for the elephant 365 days
consecutive years during this time period? after birth?

a) 0  d  2 a) 400 < w < 600
b) 2  d  3 b) 565 < w < 930
c) 4  d  8 c) 730 < w < 1,095
d) 8  d  12 d) 930 < w < 1,295

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The Ivy 37 Heart of Algebra

5. December 2018

3. March 2019 US A rental-company manager is supplying chairs and

A clothing store is having a sale on shirts and pants. During umbrellas for a pool party that 90 guests will attend. The
the sale, the cost of each shirt is $15 and the cost of each 2
pair of pants is $25. Geoff can spend at most $120 at the manager will provide enough chairs for at least 3 of the
store. If Geoff buys s shirts and p pairs of pants, which of guests and umbrellas of the where
the following must be true? for at most 2 guests,
3
a) 15s + 25p 120 each umbrella shades 3 guests. Which of the following
b) 15s + 25p 120
c) 25s + 15p 120 systems represents the number of chairs, c, and the number
d) 25s + 15p 120
of umbrellas, u, that the manager will provide for the party?

a) c ≤ 60

u ≥ 20

b) c ≥ 60

u ≥ 20

c) c ≤ 60

u ≤ 20

d) c ≥ 60

u ≤ 20

4. April 2019 US 6. April 2019 US

During an ice age, the average annual global temperature Jonathan needs to earn at least $175 next week and can work
was at least 4 degrees Celsius lower than the modern at most 20 hours. He earns $10 per hour at his lawn service
average. If the average annual temperature of an ice age is y job and $8 per hour at his job at the gym. Which of the
degrees Celsius and the modern average annual temperature following systems of inequalities represents this situation in
is x degrees Celsius, which of the following must be true? terms of the number of hours he will work at his lawn
service job, l, and the number of hours he will work at his
a) y = x – 4 job at the gym, g, next week?
b) y  x + 4
c) y  x – 4 a) 10l + 8g  175
d) y  x – 4 l + g  20

b) 10l + 8g  175
l + g  20

c) 10l + 8g  175
l + g  20

d) 10l + 8g  20
l + g  175

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Heart of Algebra 38 The Ivy

7. May 2019 US 9. May 2018 US

Terrence’s car contains gallons of fuel. He plans to drive Emma mows grass at a constant rate of 1.5 acres per hour.
the car m miles using the fuel currently in the car. If the car She mowed 2 acres before lunch and plans to spend t hours
can drive 20 miles per gallon of fuel, which inequality gives mowing after lunch. If Emma wants to mow at least 8 acres
the possible values of m? of grass today, which of the following inequalities best
represents this situation?
a) m  (8)(20)
b) m  (8)(20) a) 1.5t  8
c) 8  20m b) 1.5t – 2  8
d) 8  20m c) 1.5t + 2  8
d) 2t + 1.5  8

8. 10.

It takes 40 ink cartridges and 200 pages to print a book, and A sundae requires 3 ice-cream scoops and 4 strawberries,
it takes 30 ink cartridges and 80 pages to print a magazine. and a milkshake requires 2 ice-cream scoops and 6
Sarah wants to print books and magazines with at most 300 strawberries. Ramses wants to make sundaes and milkshakes
ink cartridges and 1200 pages. Let's form a system of with at most 25 ice-cream scoops and 37 strawberries. Let's
inequalities to represent Sarah's conditions. Let B denote the form a system of inequalities to represent Ramses's
number of books she prints and M the number of magazines conditions. Let S denote the number of sundaes he makes
she prints. Which of the following systems of inequalities and M the number of milkshakes he makes. Which system of
best describes this situation? inequalities best describes this situation?

a) 40B + 30M  300 a) 3S + 2M  25
200B + 80M  1200 6S + 4M  37

b) 40B + 30M  300 b) 3S + 2M  25
200B + 80M  1200 4S + 6M  37

c) 40B + 30M  300 c) 2S + 3M  25
200B + 80M  1200 4S + 6M  37

d) 40B + 30M  300 d) 3S + 2M  25
200B + 80M  1200 4S + 6M  37

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The Ivy 39 Heart of Algebra

11. 13.

Each morning, John jogs at 6 miles per hour and rides a bike A distributor ships DVDs to several stores. The shipping
boxes contain several DVDs (in their cases) plus a layer of
at 12 miles per hour. His goal is to jog and ride his bike a padding at each end of the box. The DVD cases and layers of
padding can be arranged neatly inside each box. Each DVD
total of at least 9 miles in less than 1 hour. If John jogs j case is 14 mm (millimeters) thick, each layer of padding is 10
mm thick, and the length of the interior of the box is 132 mm.
miles and rides his bike b miles, which of the following If d represents the number of DVDs that the distributor can fit
into one box with two layers of padding, which of the
systems of inequalities represents John’s goal?
j b following inequalities best models the situation?
a) 6 + 12  1 a) 10d + 14  132
b) 14d + 20  132
j+b9 c) 20d + 14  132
j b d) 14d + 10  132
b) 6 + 12  1

j+b<9

c) 6j + 12b  9

j+b<1

d) 6j + 12b < 1

j+b9

12. 14.
Anouk is an engineer planning sound and lighting for a free The change machine at an arcade gives change in the form of
concert in the park. The concert was advertised with a 25-cent coins (quarters) and one-dollar bills. The manager of
promise to use no more than 108 kilowatts (kW) of power. the arcade would like at least $50 worth of quarters and $100
It was determined that the main contributors to power usage, worth of one-dollar bills in the machine at all times. Also, the
speakers and floodlights, use 1.8 kW and 2.2 kW,
respectively. Anouk also must keep within her budget of manager does not allow more than $500 in the machine at any
time. What is the maximum number of quarters the manager
$3,300. The rental company is charging $75 for each would allow in the change machine?
speaker and $42 for each floodlight. Which of the following
combinations meets Anouk's requirements? a) 200
b) 400
a) 40 speakers and 30 floodlights c) 1,600
b) 12 speakers and 54 floodlights d) 2,000
c) 26 speakers and 13 floodlights
d) 38 speakers and 22 floodlights

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Heart of Algebra 40 The Ivy

15. 17.

Arianna and Hannah are conducting an experiment to test the Aaron is purchasing party hats and fake mustaches for an
difference threshold for sense of touch on the forehead. For end of the school year party. He can spend at most $30. Fake
trial 1, Hannah closes her eyes while Arianna gently touches mustaches cost $3 each, and party hats cost $2 each. If he
2 paperclip points 3 millimeters ( mm) apart on Hannah's must purchase a combination of at least 15 mustaches and
forehead. Then Hannah reports whether she could feel 1 or 2 party hats, which of the following systems of inequalities
points. For each of the next trials, Arianna and Hannah repeat best models the relationship between the number of
this process, each time with the points 2 more mm apart. mustaches, m, and party hats, p, described above?
Hannah reports feeling 1 point until the points are 15mm
a) m + p ≥ 15
apart, when she begins reporting that she feels 2 points.
Which inequality best represents the trial numbers in which 3m + 2p ≤ 30
Hannah feels 2 points? b) m + p ≤ 15

a) t > 6 3m + 2p ≥ 30
b) t < 6 c) 3m + 2p ≥ 15
c) t  7
d) t  7 m + p ≤ 30
d) 3m + 2p ≤ 15

m + p ≥ 30

16. 18.
Prachi needs between 240 and 260 total feet of fence panels. A rancher wants to build a rectangular pen for her animals.
The home improvement store has 24 short (6-foot) fence She decides that the length, l, of one side of the pen should
panels and 28 long (8-foot) fence panels in stock. Which of be at most 60 feet, the width, w, of one side of the pen
the following combinations of fence panels could she buy? should be at least 30 feet, and the perimeter of the pen
should be at most 200 feet. Which of the following systems
a) 6 short fence panels and 26 long fence panels
b) 8 short fence panels and 23 long fence panels of inequalities best models the situation described above?
c) 27 short fence panels and 12 long fence panels a) l + w ≤ 200
d) 30 short fence panels and 10 long fence panels w ≥ 30
l ≤ 60
b) l + w ≥ 200
w ≤ 30
l ≥ 60
c) l + w ≤ 100
w ≥ 30
l ≤ 60

d) 2l + 2w ≥ 200
w ≥ 30
l ≤ 60

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The Ivy 41 Heart of Algebra

19. 21.

Roberto plans to start a new job. In preparation, he decides In golf, someone who completes a course in fewer strokes
that he should spend no more than 30 hours per week on the beats someone who completes it with more strokes. Par
job and homework combined. If Roberto wants to have at means the standard number of strokes it takes to complete a
least 2 homework hours for every 1 hour at his job, what is specific course. Amateurs generally require more strokes.
the maximum number of hours that he should spend at his Ayumi averages 1.5 times par strokes per game, but her
job each week? brother Daiki averages 1.75 times par strokes per game.
To make the game more even, Ayumi adds 18 strokes to her
a) 9 hours total when she plays against Daiki.
b) 10 hours
Assuming both Ayumi and Daiki play average games, which
c) 20 hours inequality best represents the par, p, in a golf game where
d) 21 hours Ayumi beats her brother Daiki?

a) p > 41
b) p > 54
c) p > 72
d) p > 90

20.
Rodolfo wants to determine how best to benefit from his
weekly workouts. His goal is to burn at least 3300 calories
per week from jogging and playing soccer. He can burn 700
calories each hour he spends jogging and 600 calories each
hour he spends playing soccer. If Rodolfo's schedule permits
him to spend at most 3 hours per week playing soccer and at
most 5 total hours per week exercising, which of the

following exercise schedules will allow him to meet his
weekly goal?

a) 4 hours playing soccer and 1 hour jogging
b) 1 hours playing soccer and 5 hours jogging
c) 3 hours playing soccer and 2 hours jogging
d) 2 hours playing soccer and 3 hours jogging

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Heart of Algebra 42 The Ivy

Solving Linear Equations and System of Linear Equations

1. 3.
If 4x + 2 = 4, what is the value of 4x – 6?
6 = –s + 77
Given the above equation, what is the value of 1 + 5(77 – s)? a) –6
b) –4
a) –739 c) 4
b) –29 d) 8
c) 31
d) 741

2. 4.
–2(3x – 2.4) = –3(3x – 2.4) If 5 = x – 2, what is the value of (x – 2)2?
What is the solution to the equation above?
a) 5
b) 7
c) 5
d) 25

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The Ivy 43 Heart of Algebra

5. March 2019 8. March 2019 US
ax – 4(3 + 2x) = –12
3x – 5y = 17
In the equation above, a is a constant. For what value of a
ax + 15y = 11 does the equation have infinitely many solutions?
In the system of equations above, a is a constant. If the
system of equations has no solution, what is the value of a? a) –8
b) –2
a) –9 c) 2
b) –3 d) 8
c) 3
d) 9

6. March 2019 9.
4 + 5m = 4m + 1 + m
2x – 3y = 5m Which of the following best describes the solution set to the
equation shown above?
10x + ky = 25 a) The equation has exactly one solution, m = 0.
b) The equation has exactly one solution, m = 1.
In the system of equations above, k and m are constants. c) The equation has no solutions.
d) The equation has infinitely many solutions.
If the system has infinitely many solutions, what is the value
k
of m ?

a) –15
1
b) – 15

c) 1
15
d) 15

7. 10. October 2018 US
2x + 3y = 1200
y = 3x – 1 3x + 2y = 1300
1
2 y+ x = 1 Based on the system of equations above, what is the value of
5x + 5y?
In the system of equations above, if (x, y) is the solution to
x
the system, what is the value of y ?

a) 3
8
2
b) 5

c) 3
4
4
d) 3

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Heart of Algebra 44 The Ivy

11. 14.
If 4x + y = 14 and 3x + 2y = 13, then x – y = –5x – 4y = 2a
4x – 5y = 2
Which of the following choices of a will result in a system
of linear equations with exactly one solution?
a) a can be any number
b) a can be any number except 0.8
c) a can be any number except –0.8
d) a = 0.8

12. October 2018 US 15. May 2018 US
a(x + b) = 4x + 10
a(–3x – 1) + x = 7x – 2
In the equation above, a and b are constants. If the equation
has infinitely many solutions for x, what is the value of b? The equation above has no solutions, and a is a constant.

What is the value of a?
7
a) − 3

b) –2

c) 0

d) 2

13. March 2018 16. May 2018 US

Which of the following systems of equations has infinitely 5x + 2y = 22
many solutions? 4x + y = 17
In the system of equations above, what is the value of x + y?
a) 2x – y = 4 a) 5
2x + y = 4 b) 4
c) 3
b) x – 2y = 0 d) 2
2x – y = 0

c) x – 3y = 1
2x – 6y = –2

d) 3x – 4y = 2
–9x + 12y = –6

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The Ivy 45 Heart of Algebra

17. December 2017 20. May 2018 US
3ax + 4 = 6x + b 2(x + b) = ax + c

In the equation above, a and b are constants. If the equation In the equation above, a, b, and c are constants. If the
has an infinite number of solutions, what is the value of equation has infinitely many solutions, which of the
a + b? following must be equal to c?

a) 8 a) a
b) 6
c) 4 b) b
d) 2
c) 2a

d) 2b

18. December 2017

a ax + by = c
3 b c
x+ 2 y = 6 21. October 2019 US

In the system of linear equations shown above, a, b, and c x+y=2
x–y=3
are constants, and a and b are not equal to zero. Which of If (x, y) is the solution to the system of equation above, what
is the value of x?
the following is true about the system of linear equations?

a) It has no solutions.

b) It has exactly one solution.

c) It has exactly two solutions.

d) It has infinitely many solutions.

19. 22. October 2019 US

1 s = 10r + 40 2k(x – 2) = x – 2
2 3 In the equation above, k is a constant. If the equation has
(s – 3) = 5r – 2 infinitely many solutions, what is the value of k?

Consider the system of equations above. If (r, s) is the

solution to the system, then what is the value of r∙s?

a) –375

b) –345

c) There are infinitely many solutions to this system

of equations.

d) There are no solutions to this system of equations.

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Heart of Algebra 46 The Ivy

23. 26. October 2019

–x + 3 = y 2x + 5y = 5.5
–6x + 18 = 6y –2x – 4y = 10
Which of the following accurately describes all solutions to If (x, y) is the solution to the system of equations above,
the system of equations shown above? what is the value of y?
a) x = 0 and y = 3
b) x = 3 and y = 0
c) There are infinite solutions to the system.
d) There are no solutions to the system.

24. October 2019 US 27. October 2019
In the xy-plane, the graph of y = x + 3 intersects the graph of
y = 2x – 6 at the point (a, b). What is the value of a? 5x + 2y = 20
15x + ay = 80
a) 3 In the system of equations shown, a is a constant.
If the system has no solution, what is the value of a?
b) 6 a) 2

c) 9 b) 5

d) 12 c) 6

d) 20

25. October 2019 US 28.

7x – 5y = 4 y = 4.5x + 3
4x – 8y = 9 y = C(x + 2)
If (x, y) is the solution to the system of equations above, In the system of equations above, C is a constant.
what is the value of 3x + 3y? For which values of C does the system have no solution?
a) 1.5
b) 4.5
c) All real numbers
d) None of the above

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