Systems of Linear and Quadratic Equations - PHSchool.com

NY-6 Systems of Linear and
Quadratic Equations
Learning Standards for
Mathematics

A.A.11 Solve a system of Check Skills You’ll Need GO for Help Lessons 7-1, 7-2, and 10-4
one linear and one
quadratic equation in 1. Solve the system using substitution. 2. Solve the system by graphing.
two variables, where only xϭyϩ2 y ϭ 2x ϩ 3
factoring is required.
4x ϩ y ϭ 8 xϭy
A.G.9 Solve systems of linear
and quadratic equations 3. Solve x2 Ϫ 5x + 6 ϭ 0 by factoring.
graphically.

1 Solving Systems Using Graphing

In Lesson 7-1, you solved systems of linear equations graphically and algebraically.
A system of linear equations can have either one solution, no solutions, or infinitely
many solutions. In Chapter 10, you solved quadratic equations graphically and
algebraically.

In this lesson, you will study systems of linear and quadratic equations. This type of
system can have one solution, two solutions, or no solutions.

y ϭ x2 Ϫ 4 y ϭ x2 y ϭ x2 ϩ 4
y ϭ Ϫ3 yϭ0 yϭxϩ1

y yy
2

45

Ox 2 x 2
Ϫ2 Ϫ2 O 2 Ϫ2 O

2x

two solutions one solution no solutions

1 EXAMPLE Solve by Graphing

Solve the following system by graphing. y ϭ x2 ϩ x Ϫ 2 y
y ϭ Ϫx ϩ 1

Graph both equations on the same coordinate plane. (᎐3, 4)
Identify the point(s) of intersection, if any. 3

The points (Ϫ3, 4) and (1, 0) are the solutions of O (1, 0) x
the system. ᎐3 4

Quick Check 1 Solve the system by graphing. y ϭ 2x ϩ 2
y ϭ Ϫx2 Ϫ x ϩ 2

NY 752 Chapter NY New York Additional Topics

2 EXAMPLE Graphing to Count Solutions

Find the number of solutions for the system. y ϭ 2x2 ϩ 3
yϭxϩ2

Step 1 Graph both equations on the y
same coordinate plane. 6

Step 2 Identify the point(s) of intersection, if any. 3
O2
There are no points of intersection, so there is no x
solution to the system of equations.

᎐4

Quick Check 2 Find the number of solutions for each system.

a. y ϭ x Ϫ 4 b. y ϭ x2 Ϫ 6x ϩ 10
y ϭ 2x2 ϩ x yϭ1

12 Solving System Using Algebraic Methods

In Lesson 7-3, you solved linear systems using elimination. The same technique can
be applied to systems of linear and quadratic equations.

3 EXAMPLE Using Elimination

Solve the following system of equations: y ϭ x2 Ϫ 11x Ϫ 36
y ϭ Ϫ12x ϩ 36

Step 1 Eliminate y. Subtract the two equations.
y ϭ x2 Ϫ 11x Ϫ 36 Subtraction Property of Equality

Ϫ (y ϭ Ϫ 12x ϩ 36)
0 ϭ x2 ϩ x Ϫ 72

Step 2 Factor and solve for x
0 ϭ x2 ϩ x Ϫ 72

0 ϭ (x ϩ 9)(x Ϫ 8) Factor.
x ϩ 9 ϭ 0 or x Ϫ 8 ϭ 0 Zero-Product Property

x ϭ Ϫ9 or xϭ8

Step 3 Find the corresponding y values. Use either equation.

y ϭ x2 Ϫ 11x Ϫ 36 y ϭ x2 Ϫ 11x Ϫ 36

y ϭ (Ϫ9)2 Ϫ 11(Ϫ9) Ϫ 36 y ϭ (8)2 Ϫ 11(8) Ϫ 36

y ϭ 81 ϩ 99 Ϫ 36 y ϭ 64 Ϫ 88 Ϫ 36

y ϭ 144 y ϭ Ϫ60

The solutions are (Ϫ9, 144) and (8, Ϫ60).

Quick Check 3 Solve the system using elimination. y ϭ x2 ϩ 4x Ϫ 1

y ϭ 3x ϩ 1

See back
of book.

Lesson NY-6 Systems of Linear and Quadratic Equations NY 753

4 EXAMPLE Using Substitution

Solve the following system of equations: y ϭ x2 Ϫ 6x ϩ 9 and y ϩ x ϭ 5.

Step 1 Solve y ϩ x ϭ 5 for y. Subtract x from both sides.
yϩxϪxϭ5Ϫx
yϭ5Ϫx

Step 2 Write a single equation containing only one variable.

y ϭ x2 Ϫ 6x ϩ 9

5 Ϫ x ϭ x2 Ϫ 6x ϩ 9 Substitute 5 ؊ x for y.

5 Ϫ x Ϫ (5 Ϫ x) ϭ x2 Ϫ 6x ϩ 9 Ϫ (5 Ϫ x) Subtract 5 ؊ x from both sides.

0 ϭ x2 Ϫ 5x ϩ 4

Step 3 Factor and solve for x.

0 ϭ (x Ϫ 4)(x Ϫ 1) Factor.
Zero-Product Property
x Ϫ 4 ϭ 0 or x Ϫ 1 ϭ 0

x ϭ 4 or xϭ1

Step 4 Find the corresponding y-values. Use either equation.

y ϭ Ϫx2 ϩ 4x ϩ 1 y ϭ Ϫx2 ϩ 4x ϩ 1

ϭ Ϫ(42) ϩ 4(4) ϩ 1 ϭ Ϫ(12) ϩ 4(1) ϩ 1

ϭ1 ϭ4

The solutions of the system are (4, 1) and (1, 4).

Quick Check 4 Solve the system using substitution. y Ϫ 30 ϭ 12x
y ϭ x2 ϩ 11x Ϫ 12

In Lesson 10-7, you used the discriminant to find the number of solutions of a
quadratic equation. With systems of linear and quadratic equations you can also
use the discriminant once you eliminate a variable.

5 EXAMPLE Using the Discriminant to Count Solutions

At how many points do the graphs of y ϭ 2 and y ϭ x2 ϩ 4x ϩ 7 intersect?

Step 1 Eliminate y from the system. Write the resulting equation in
standard form.

y ϭ x2 ϩ 4x ϩ 7

Ϫ (y ϭ 2) Subtract the two equations.

0 ϭ x2 ϩ 4x ϩ 5 Subtraction Property of Equality

Step 2 Determine whether the discriminant, b2 Ϫ 4ac, is positive, 0, or negative.

b2 Ϫ 4ac ϭ 42 Ϫ 4(1)(5) Evaluate the discriminant.

ϭ 16 Ϫ 20 Use a ‫ ؍‬1, b ‫ ؍‬4, and c ‫ ؍‬5.

ϭ Ϫ4

Since the discriminant is Ϫ4, there are no solutions. The graphs do not intersect.

Quick Check 5 At how many points do the graphs of y ϭ x2 Ϫ 2 and y ϭ x ϩ 5 intersect?

NY 754 Chapter NY New York Additional Topics

6 EXAMPLE Solve Using a Graphing Calculator

Solve the system of equations y ϭ Ϫx2 ϩ 4x ϩ 1 and y ϭ Ϫx ϩ 5 using a graphing
calculator.

Step 1 Step 2 Step 3

Y1=-X2+4X+1

First curve? Y=1 Intersection Y=4
X=0 X=1

Enter y ϭ Ϫx2 ϩ 4x ϩ 1 Use the feature. Move the cursor close to
and y ϭ Ϫx ϩ 5 into
Y1 and Y2. Press Select 5: Intersect. a point of intersection.

to display Press three
the system.
times to find the point of
intersection.

Step 4 Repeat Steps 2 and 3 to find the second
intersection point.

The solutions of the system are (1, 4) and (4, 1). Intersection Y=1
X=4
6 Solve the system using a graphing calculator. y ϭ x2 Ϫ 2
Quick Check y ϭ Ϫx

EXERCISES For more exercises, see Extra Skill and Word Problem Practice.

Practice and Problem Solving

A Practice by Example Solve each system by graphing. Find the number of solutions for each system.

Examples 1 and 2 1. y ϭ x2 ϩ 1 2. y ϭ x2 ϩ 4 3. y ϭ x2 Ϫ 5x Ϫ 4
(pages NY 752 and NY 753) yϭxϩ1 y ϭ 4x y ϭ Ϫ2x

GO for 4. y ϭ x2 ϩ 2x ϩ 4 5. y ϭ x2 ϩ 2x ϩ 5 6. y ϭ 3x ϩ 4
Help yϭxϩ1 y ϭ Ϫ2x ϩ 1 y ϭ Ϫx2

Example 3 Solve each system using elimination.
(page NY 753)
7. y ϭ Ϫx ϩ 3 8. y ϭ x2 9. y ϭ Ϫx Ϫ 7
y ϭ x2 ϩ 1 yϭxϩ2 y ϭ x2 Ϫ 4x Ϫ 5

10. y ϭ x2 ϩ 11 11. y ϭ 5x Ϫ 20 12. y ϭ x2 Ϫ x Ϫ 90
y ϭ Ϫ12x y ϭ x2 Ϫ 5x ϩ 5 y ϭ x ϩ 30

Example 4 Solve each system using substitution.
(page NY 754)
13. y ϭ x2 Ϫ 2x Ϫ 6 14. y ϭ 3x Ϫ 20 15. y ϭ x2 ϩ 7x ϩ 100
y ϭ 4x ϩ 10 y ϭ Ϫx2 ϩ 34 y ϩ 10x ϭ 30

16. Ϫx2 Ϫ x ϩ 19 ϭ y 17. 3x Ϫ y ϭ Ϫ 2 18. y ϭ 3x2 ϩ 21x Ϫ 5
x ϭ y ϩ 80 2x2 ϭ y Ϫ10x ϩ y ϭ Ϫ1

Lesson NY-6 Systems of Linear and Quadratic Equations NY 755

Example 5 Use the discriminant to find the number of solutions for each system.
(page NY 754)
19. y ϭ x2 Ϫ 5x Ϫ 8 20. y ϭ Ϫx2 Ϫ 3 21. y ϭ Ϫ3x Ϫ 6
Example 6 yϭx y ϭ 9 ϩ 2x y ϭ 2x2 Ϫ 7x
(page NY 755)
22. y ϭ 25x2 Ϫ 9x ϩ 2 23. y ϭ Ϫx2 Ϫ 4x ϩ 9 24. 4x2 ϩ 20x ϩ 29 ϭ y
B Apply Your Skills y ϩ 2 ϭ 11x
y ϩ 5x ϭ Ϫ7 8x ϩ y ϩ 20 ϭ 0

Solve each system using a graphing calculator.

25. y ϭ x2 Ϫ 2x Ϫ 2 26. y ϭ Ϫx2 ϩ 2 27. y ϭ x Ϫ 5
y ϭ Ϫ2x ϩ 2 y ϭ 4 Ϫ 0.5x y ϭ x2 Ϫ 6x ϩ 5

28. y ϭ Ϫ0.5x2 Ϫ 2x ϩ 1 29. y ϭ 2x2 Ϫ 24x ϩ 76 30. Ϫx2 Ϫ 8x Ϫ 15 ϭ y

y ϩ 3 ϭ Ϫx y ϩ 7 ϭ 11 Ϫx ϩ y ϭ 3

31. Critical Thinking The graph at the right y
shows a quadratic function and the 4
linear function y ϭ d.
a. If the linear function were changed to 2 x
y ϭ d ϩ 3, how many solutions would the 246
system have? O
b. If the linear function were changed to ᎐4 ᎐2
y ϭ d Ϫ 5, how many solutions would the
system have? Ϫ3

Solve each system using either elimination or substitution.

32. y ϭ 2x2 ϩ 13x 33. y ϭ Ϫ8x 34. y ϭ x2 ϩ 9x Ϫ 91
y ϭ Ϫ9 Ϫ 6x y ϭ 1 ϩ 16x2 y
xϭ 3
35. y ϩ 20x ϭ 39 36. y ϭ x2 Ϫ 12x Ϫ 20
15 ϩ 4x2 ϩ 9x ϭ y y ϭ 25(4 Ϫ x) 37. 5x2 ϩ 14x ϩ 1 ϭ y

Ϫ12 ϩ y ϩ 40x ϭ 0

38. Graphing Calculator The screen at the right shows X Y1 Y2
the y- and x-values for the system y ϭ x2 Ϫ 6x ϩ 8
and y ϭ x Ϫ 1. Use the table to find the -1 15 -3
solutions of the system. 0 8 -2
1 3 -1
39. Writing Explain why a system of linear and 2 0 0
quadratic equations cannot have an infinite number 3 -1 1
of solutions. 4 0 2
5 3 3

X= -1

Use substitution and the quadratic formula to find the solutions of each system.
Round your answers to the nearest hundredth.

40. y ϭ 2x2 ϩ 4x Ϫ 1 41. 2y ϩ 4 ϭ x 42. 3x ϭ y Ϫ 7
Ϫ5x ϩ y ϭ 5 y ϩ x2 ϭ 4 y ϭ 6x2 Ϫ 4x ϩ 1

43. The graph at the right shows the system y ϭ x2 Ϫ 5 y
and y ϭ x. Find the values of x such that the y-values 12
on the parabola are 10 units greater than the 8
corresponding y-values on the line. Round your 4
answers to the nearest hundredths.

44. Critical Thinking Solve the system y ϭ x2 ϩ x ϩ 25 O 4 x
and y ϭ x using substitution. How can you tell that ᎐8 ᎐4 8
the system has no solutions without using graphing,
the discriminant, or the quadratic formula? ᎐6

NY 756 Chapter NY New York Additional Topics

C Challenge 45. Geometry The figures below show rectangles that are centered on the y-axis

with bases on the x-axis and upper vertices defined by the function
y = –0.3x2 ϩ 4. Find the area of each rectangle. Round to the nearest hundredth.

a. y b. 6 y
6

44

2 x 2 x
24 24
᎐4 ᎐2 ᎐4 ᎐2
᎐2 ᎐2

c. Find the x- and y-coordinates of the vertices of the square constructed in the
same manner.

d. Find the area of the square. Round to the nearest hundredth.

REGENTS

Test Prep

Multiple Choice 46. Which coordinate pair is a solution to the following system?
Short Response
y ‫ ؍‬x2 ؉ 2x ؊ 2

y ‫ ؍‬x ؉ 10

A. (4, 14) B. (3, 13) C. (2, 12) D. (؊3, 7)

47. The graph at the right shows the system
y ‫ ؍‬x ؉ 4 and y ‫ ؍‬؊x2 ؉ x. How many solutions
does the system have?
F. one solution
G. two solutions
H. no solutions
J. cannot be determined

48. Use the discriminant to determine the y ‫ ؍‬49x2 ؊ 2x ؉ 34
number of solutions of the system. y ؉ 30 ‫ ؍‬100x

49. Solve the system using substitution and x2 ؉ 3x ؊ 23 ‫ ؍‬y
factoring. Show your work. y
5 - 5 = x

Mixed Review

Lesson NY-5 Tell whether each correlation is a causal relationship. Justify your answer.
Lesson NY-4
50. Hours of computer use and television viewing have a negative correlation.
Is this a causation?

51. The number of ice cream trucks in a town on a given day and the high
temperature have a positive correlation. Is this a causation?

Find each union or intersection. Let A ‫{ ؍‬1, 3, 6}, B ‫{ ؍‬2, 6, 8}, C ‫{ ؍‬x | x is an even
number less than 6}, and D ‫{ ؍‬x | x is a multiple of 3 less than 12}.

52. A e B 53. A f B 54. A f D 55. C f D

56. B f C 57. D e C 58. D e A 59. A f C

Lesson NY-6 Systems of Linear and Quadratic Equations NY 757