Algebra Textbook Volume 1 Student Edition wc

Practice 6.15 B Name _________________

1) In each case below, y varies directly with x. Determine the values of the constant of

variation.

A. y = 42 when x = 6 B. y = 15 when x = 30

C. y = 56 when x = 7 D. y = 2 when x = 8

2) Do the following relationships show direct variation? If so, state the relationship and
the constant of variation.
(A) (B)

xy xy
4 16 16 8
6 24 18 9
8 32 20 10

Relationship: Relationship:

Constant of variation (k) Constant of variation (k)

(C) (D)
xy xy
28 39
3 24 6 18
4 48 9 27
Relationship:
Relationship:
Constant of variation (k)
Constant of variation (k)
102 | P a g e

6.15 B (cont’d) Name _________________
3) In each case, the value of y varies directly with x.
A) y = 18 when x = 3. Find y when x = 5

B) y = 5 when x = 10. Find y when x = 12

C) y = 36 when x = 9. Find y when x = 12

D) y = 50 when x = 5. Find y when x = 3

103 | P a g e

Cloze Responses

6.1A
Observing these points on the graph, it appears that they _form___ _a_ __straight__
__line__. Draw on the graph, extending the graph in both directions.
The previous two examples demonstrate that ___ the coordinates of the points on
the line work in the equation and those of points not on the line do
not work in the equation_______.

6.2A

Equations with x and y values define sets of ordered pairs. Some equations, called __linear_
equations, give sets of ordered pairs that correspond to straight lines when graphed.

There are several aspects of a line that can be used to describe and then graph a line. The three
most common are the __slope__, the x-intercept, and the y-intercept.

By examining the lines drawn above, it can be observed that, in the case of line A, for each step

to the right, the line goes up by one (+1). Therefore, its slope is 1 ( 1 1 ). In the case of
ℎ

line B, for each step to the right the line goes up 2. This line, therefore, has a slope of 2

(12ℎ ). The third line (line C) has a slope of _3__. For each step to the right the line goes

up _3_ _steps_. Draw a line with a slope of 4 on the same graph and label it D. These slopes

are all positive because, as a step to the right is taken, the y-coordinate increases.

By examining the lines drawn above, it can be observed that in the case of line E, for each step

to the right, the line goes down by one. Therefore, its slope is -1, which means(11 ). In
ℎ

the case of line F, for each step to the right, the line goes down 2. Therefore, this line has a

slope of -2, which means (21ℎ ). The third line, line G, would have a slope of __-3____

( ). Draw a line with a slope of -4 on the same graph and label it H.


104 | P a g e

These slopes are all negative because, as a step to the right is taken, the y-coordinate
decreases.

6.4A

As seen earlier, the slope of a line can be determined from the graph of the lines

(# ) . The slope of a line can also be calculated if _two_ points on the line are
1 ℎ ℎ

known. The __change__in the y-coordinate is divided by the change in the x-coordinate.

6.5A
Straight lines are also described by specifying their x-intercepts and y-intercepts. The x-
intercept is the point where the line crosses the __x-axis__ (x, 0) and, as expected, the y-
intercept is the point where the line _crosses_ the y-axis (0, y).
When given a graph, identifying the intercepts is very straight-forward. Simply, locate the x-
and y- _axes_. Then identify the two points where the line crosses them. If there is no
intercept (point where the line crosses), write “no intercept.”

6.6A
The most useful form for the equation of a straight line is the slope-intercept form. As the name
suggests, the slope-intercept form is useful because the __slope_ and the y-intercept of the
line are given directly. In the equation the letter _b__ is used to represent the y-intercept.
(There is no variable routinely used for the x-intercept.)

6.7A
It is also easy to write the _equation__ for a graphed line in slope-intercept form. First,
locate the y-axis and find the y-intercept (__0___ , _b___). Then find any other point on the
line close to the y- intercept. Use the two points to determine the ___slope__ (m). Then
write the equation of the line in slope-intercept form: y = mx+b

105 | P a g e

6.8
Another commonly used form for the equation of a line is called __standard___
_form____________.
Standard form for the equation of a line is ___Ax + By = C__. A, B, and C must be integers
and A must be positive. Unlike the slope-intercept form for the equation of a line (__y = mx
+ b_) where the slope of the line (__m___) and its y-intercept (_b__) are given directly, the
values of A, B, and C given in the equation of a line written in standard form do not directly
refer to any actual aspects of the line.

The standard form for the equation of a line is particularly useful in graphing some lines. In
these cases, it is very easy to use the equation in the standard form to determine the x- and y-
intercepts of the line. These can be located on the graph and joined by a line. Thus, the
equation is easily graphed.

Determining the intercepts from the equation in standard form.
The x-intercept: The coordinates of the x-intercept of a line will be (_a number___,
__0____). To find the x-intercept when the equation of the line is given in standard form,
simply substitute zero for y and solve the equation for x. An easy way to accomplish this is to
use a finger to cover By in Ax + By = C since By = B0 = 0 and then solve for x. This obviously
works well if C is divisible by A.

The y-intercept: The coordinates of the y-intercept of a line will be (_0__, __a number___).
To find the y-intercept when the equation of the line is given in standard form, simply
substitute zero for x and solve the equation for y. An easy way to accomplish this is to use a
finger to cover Ax in Ax + By = C since Ax = A0 = 0 and then solve for y. This obviously works
well if C is divisible by B.

106 | P a g e

6.10A
Often linear equations are originally written in ___standard form_ (__Ax+By=C_) and
must be rearranged to ___slope__- _intercept__ form (__y=mx+b ____). Changing from
standard form to slope-intercept form is a two-step process. Essentially, the change is made by
solving the equation for y.

6.10A (cont’d)
Converting from slope-intercept form (__y=mx+b__) to standard form (__Ax+By=C ___)

6.11A
Another form for the equation of a straight line is called the point-slope form. Given the
__slope__ (m) and a __point_ (x1, y1), on the line, the point-slope form for the equation of
the line is:

6.12A
Three forms of linear equations have been introduced thus far: the _slope - __intercept_ form
(y=mx+b), the __standard__ form (Ax + By = C), and the __point___-_slope__ form [(x-x1)
=m(y-y1)]. In this section, some information is given regarding the line, and from that
information, the _equation__ of the line in the form requested must be found. In general, these
problems will require finding the __slope_ of the line and then either __one_ point on the line or
the y-intercept. The final step may require rewriting the equation in the requested form.

6.13 A
Parallel lines are lines that never meet or _intersect___. The points on each line are always
equidistant from each other. See Graph A. Parallel lines always share the _same__ slope. In
this case, the slope of both lines is 3. Perpendicular lines, however, do intersect, but only at a
_right__ angle (900). See Graph B. The slopes of perpendicular lines are always the opposite,
_reciprocal_ of each other. If the slope of one line is 3, then the slope of the other line
would be − 31. Basically, you flip the number (reciprocal) and change the sign (opposite).

107 | P a g e

6.14A
In Chapter 5, relations and functions were covered, including the requirements for a
__function_: Every x-value is paired with one and only __one_ y –value. This last section of
Chapter 6 describes a simple test to determine if a set of _ordered_ pairs (x1, y1)(x2, y2) forms a
linear function.
To test to see if a set of ordered pairs form a linear function, determine the pattern of change
for both the x- and y- values. If it is observed that, as the x-values undergo aconstant_
change, the y-values also undergo a constant change, then the ordered pairs are linear.
6.15A
There are cases when two variables usually x and y are related in such a way that whatever
happens to x will happen to y. For example, if the value of x doubles, the corresponding value of
y _doubles_. If the value of x is cut in half, the corresponding value of y will also be cut in
half. In other words, y varies directly with x. This, relationship between the variables is called a
__direct____ ___variation__.

A direct variation is described with an equation in the form y = kx, where k has a constant value
called the ___constant__ ___of__ _variation__, and x and y values are the ordered
pairs. Consider the example below:

108 | P a g e

Chapter 6 Review

y

Find the slope of each line: 4 5
1) _______________ 21 x

2) _______________ 3 6
3) _______________
4) _______________ 109 | P a g e
5) _______________
6) _______________

Given two points on the
line, find the slope of the
line:
_______________ 7) (2, 8) (5, 6)

_______________ 8) (−7, −12) (−5, −2)

_______________ 9) (3, −8) (3, 2)

_______________ 10) (2, −10) (6, −7)

_______________ 11) (5, −1) (6, −1)

_______________ 12) (7, −5) (3, 3)

C6 Review (cont’d) Name ________________________

Determine the intercepts of the lines whose equations are given. Then graph and label the lines

using those intercepts. Graph both lines on the same axis.

y

13) 7 − 14 = −28
x-intercept: ____________
y-intercept: ____________

x

14) 4 − 6 = 24
x-intercept: ____________
y-intercept: ____________

Determine the equation in slope-intercept form for each line graphed.

15) ___________________ y 17
15

16) ___________________ 16

17) ___________________ 18
19 x

18) ___________________

19) ___________________ 20

20) ___________________

110 | P a g e

C6 Review (cont’d) Name ________________________
y
Graph and label the lines below on the same axes:
x
21) = 2 + 4
3

22) = −4

23) = −3 + 5

24) = 2 − 2
5

25) = 3

26) = − 1 − 4
2

27) Arnie needs to buy movie tickets ($12 each) and amusement park passes ($10 each) to use
as prizes at his fundraiser. He has $120 to spend. Graph the line that represents the
numbers of movie tickets and amusement park passes that Arnie can buy.

A. Let x = _______________________________________________________________
Let y = _______________________________________________________________

B. Equation: ________________________________
C. Intercepts: x-intercept: (______, ______)

y-intercept: (______, ______)
D. Graph the equation:
E. Using the graph, predict the number of amusement

Park passes Arnie can buy along with 5 movie tickets.

111 | P a g e

C6 Review (cont’d) Name ________________________

Write each equation below in slope-intercept form.

29) 4 + 7 = −28 ___________________________________

30) 6 − 5 = 30 ___________________________________

Write each equation below in standard form.

31) = − 1 − 8 ___________________________________
2

32) = 1 + 7 ___________________________________
4

Write the equation of each line described below in point-slope form.
33) slope −2 and contains point (−3, 5) ___________________________________
34) slope 4 and contains point (2, −7) ___________________________________

35) The point-slope form for the equation of a line is − 9 = 1 ( + 4)
2

The slope of this line is ____________________________________

One point on the line is _________________

36) The point-slope form for the equation of a line is − 2 = 3 ( + 7)
4

The slope of this line is ____________________________________

One point on the line is _________________

112 | P a g e

C6 Review (cont’d) Name ________________________

Find the equation of each line described below in slope-intercept form and standard form:
37) m = −4 and line goes through (−3, 6)

Slope-intercept form: _________________________________
Standard form: ________________________________

38) m = and line goes through (−8, 1)
Equation: _________________________________

39) m = 2 and line goes through (5, −3)
Slope-intercept form: _________________________________
Standard form: ________________________________

40) m = 0 and line goes through (2, 9)
Equation: _________________________________

41) m= 1 and line goes through (−8, 5)
4

Slope-intercept form: _________________________________

Standard form: ________________________________

42) m= 1 and line goes through (20, −10)
5

Slope-intercept form: _________________________________

Standard form: ________________________________

113 | P a g e

C6 Review (cont’d) Name ________________________

43) line goes through (−3, 2) and (−9, 6)
Slope-intercept form: _________________________________
Standard form: ________________________________

44) line goes through (7, 15) and (2, 0)
Slope-intercept form: _________________________________
Standard form: ________________________________

45) What is the slope of the line parallel to the line with equation = −5 + 2 ?
Slope (m): _________________________________

46) What is the slope of the line perpendicular to the line with equation = − 1 − 3?
2

Slope (m): _________________________________

47) What is the slope of the line perpendicular to the line with equation = −3 + 5?

Slope (m): _________________________________

48) line goes through (4, −3) and is parallel to the line with equation = 1 − 8
2

Slope-intercept form: _________________________________

Standard form: ________________________________

114 | P a g e

C6 Review (cont’d) Name ________________________

49) line goes through (5, 9) and is parallel to the line with equation = −4 − 1
Slope-intercept form: _________________________________
Standard form: ________________________________

50) line goes through (2, −6) and is perpendicular to the line with equation = −2 + 3
Slope-intercept form: _________________________________
Standard form: ________________________________

51) line goes through (6, −7) and is perpendicular to the line with equation = 1 + 11
3

Slope-intercept form: _________________________________

Standard form: ________________________________

52) Do the following sets of ordered pairs form functions?

A. B. C. D.
xy xy xy xy
1 16 23 45 13
4 14 46 8 10 25
7 12 89 12 15 38
10 10 14 12 16 20 4 12
yes no
yes no yes no yes no

52) The value of y varies directly with x.
A. If = 30 when = 5, find y when = 8: = _________________

B. If = 6 when = 12, find y when = 24: = _________________

115 | P a g e

C6 Review (cont’d) Name ________________________

C. If = 45 when = 9, find y when = 12: = _________________

D. If = 5 when = 15, find y when = 27: = _________________

116 | P a g e

Key Vocabulary Chapter Seven

Introduction 1|Page
• forms of linear equations

o standard set
o slope-intercept
o point-slope
• ordered pairs
• line
• intersect
• intersection point
• system of equations
• two equations, two
unknowns

7.2 A
• substitution

7.3 A
• elimination

7.4 A
• Cramer’s Rule

7.6 A
• linear inequality
• infinite set

7.7 A
• linear programming
• constraints
• feasibility region
• corner points
• profit equation

Chapter Seven: Systems of Equations
Section One – Solution Methods & Graphing
Introduction to Solving Systems of Equations
Graphing Method....................................................................................................................... 7.1A
Quiz ................................................................................................................................ 7.1D
Substitution Method.................................................................................................................. 7.2A
Quiz ................................................................................................................................ 7.2D
Elimination Method ................................................................................................................... 7.3A
Quiz .................................................................................................................................7.3E
Cramer’s Rule Method............................................................................................................... 7.4A
Quiz .................................................................................................................................7.4E
Graphing Inequalities................................................................................................................. 7.5A
Quiz .................................................................................................................................7.5E
Section Two – Systems of Linear Inequalities & Graphing
Solving Systems of Linear Inequalities....................................................................................... 7.6A
Quiz ................................................................................................................................ 7.6D
Linear Programming .................................................................................................................. 7.7A
Linear Programming Exploration (Real World Application) (Appendix)........................ 7.7B
Section Three – Review: Solution Methods (cont.)
Review of four methods............................................................................................................. 7.8A

Cloze Responses
Chapter Review & Chapter Test

2|Page

Chapter 7 Introduction

Chapter 7 encompasses solving two equations which contain two variables. These equations can
be written in different forms, the three most common being:

Standard form: _________________________________
Slope-intercept form: ___________________________
Point-slope form: _______________________________

These were introduced in the last chapter. In this chapter, the focus is on solving sets of these
equations. The slope-intercept form will be used at the beginning of the chapter.

An equation in the form y=mx+b such as y=2x+3 describes the relationship between x- and y-
values given in ordered pairs. There are an infinite number of ________________ ___________
have x- and y- values that fit the equation y=2x+3. If these ordered pairs were graphed, they
would give a straight line that goes on forever in both directions. Every ordered pair that works
in this equation is a point on that line. The ordered pairs for points not on that line will
__________ work in that equation. The ordered pairs whose x- and y- values are related by the
equation y=3x+1 will also result in a __________________ _____________________ when
graphed. These two lines intersect- they cross each other. They have one point in common, the
point where they cross. The coordinates of that point will work in ________________equations.

Students: Graph y=2x+3 and y = − 1 + 8 and identify the point where they cross:
2

3|Page

Introduction (cont.) Name_______________________________
This point will work in both equations:
Point: (2,7)

y=2x+3 y=− 1 + 8
7=22+3 2
7=7
7=− 1 (2) + 8
2

7=−1 + 8

7= 7

A system of equations is simply a group of __________________. When asked to solve or find

the solution for a system of equations, in this case the system of equations is y=2x+3 and

y=− 1 + 8, the object is to identify the one pair of x- and y- values that will work in
2

_____________ equations. In this case the solution is (2,7).

Solving systems of equations can be done in several different ways. Those covered in this text
are:

1) Graphing
2) Substitution
3) Elimination
4) Cramer’s Rule

4|Page