Name
Class
5.2
Date
Solving Linear-Quadratic
Systems
Essential Question: How can you solve a system composed of a linear equation in two
variables and a quadratic equation in two variables?
Resource
Locker
A2.3.C: Solve, algebraically, systems of two equations in two variables consisting of a linear
equation and a quadratic equation. Also A2.3.A, A2.3.D
Investigating Intersections of Lines
and Graphs of Quadratic Equations
Explore
There are many real-world situations that can be modeled by linear or quadratic functions. What happens when the
two situations overlap? Examine graphs of linear functions and quadratic functions and determine the ways they can
intersect.

Examine the two graphs below to consider the ways a line could intersect the parabola.
8
y
8
y
4
4
x
x
-8
-4
0
4
-8
8
-4
-4
© Houghton Mifflin Harcourt Publishing Company
4
8
-4
-8
-8
B
0
Sketch three graphs of a line and a parabola: one in which they intersect in one point, one in
which they intersect in two points, and one in which they do not intersect.
8
y
8
4
y
8
4
4
x
x
-8
-4
0
-4
-8

4
8
-8
-4
0
4
8
-4
-8
x
-8
-4
0
4
8
-4
-8
A constant linear function and a quadratic function can intersect at
Module 5
y
255
points.
Lesson 2
Reflect
1.
If a line intersects a circle at one point, what is the relationship between the line and the radius of the circle
at that point?
2.
Discussion If a line is not horizontal, at how many points can it intersect a parabola?
Explain 1
Solving Linear-Quadratic Systems Graphically
Graph each equation by hand and find the set of points where the two graphs intersect.
Solve the given linear-quadratic system graphically.
Example 1

⎧2x - y = 3
⎨
2
⎩ y + 6 = 2(x + 1)
Plot the line and the parabola.
Solve each equation for y.
8
2x - y = 3
y = 2x - 3
4
y
x
-8
y + 6 = 2(x + 1)
2
y = 2(x + 1) - 6
2
-4
0
4
8
Find the approximate points of intersection: Estimating from the graph, the intersection points appear to
be near (-1.5, -5.5) and (0.5, -2.5).
(
)
_
_
-1 + √ 3
-1 - √3
The exact solutions (which can be found algebraically) are _______
, -√3 - 4 and ( _______
, √3 - 4 ),
2
2
or about (-1.37, -5.73) and (0.37, -2.27).
⎩
_
Plot the line and the parabola on the axes provided.
2
2
Solve each equation for y.
8
3x + y = 4.5
4
y
x
y=
-8
y=
-4
0
4
8
-4
-8
Find the approximate point(s) of intersection:
.
Note that checking these coordinates in the original system shows that this is an exact solution.
Module 5
256
Lesson 2
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⎧ 3x + y = 4.5
B ⎨ _1 ( )
y=
x- 3
_
Your Turn
Solve the given linear-quadratic system graphically.
⎧y + 3x = 0
3. ⎨
2
⎩ y - 6 = -3x
8
4.
1 (x - 3) 2
⎧y + 1 = _
⎨ x - y = 26
⎩
y
8
4
y
4
x
-8
-4
0
4
x
-8
8
-4
-4
-8
Explain 2
0
4
8
-4
-8
Solving Linear-Quadratic Systems Algebraically
Use algebra to find the solution. Use substitution or elimination.
Example 2

Solve the given linear-quadratic system algebraically.
⎧3x - y = 7
⎨
⎩ y + 4 = 2(x + 5)
2
7 + y = 3x
2
4 + y = 2(x + 5)
Solve this system using elimination.
First line up the terms.
7 + y = 3x
(―――――――― )
3 = 3x - 2(x + 5)
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Subtract the second equation from
the first to eliminate the y variable.
- 4 + y = 2(x + 5)
2
2
3 = 3x - 2(x + 5)
Solve the resulting equation for x
using the quadratic formula.
2
3 = 3x - 2(x 2 + 10x + 25)
3 = 3x - 2x 2 - 20x - 50
0 = -2x 2 - 17x - 53
2x 2 + 17x + 53 = 0
――――――
-17 ± 17 2 - 4 ⋅ 2 ⋅ 53
x = ___
2⋅ 2
――――
-17 ± 289 - 424
= __
4
――
There is no real number equivalent to -135 , so the system
has no solution.
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257
――
-17 ± -135
= __
4
Lesson 2
1
_
 ⎧y = ​ 4 ​ ​(x - 3)​ ​
     
​
⎨  ​     ​
⎩ 3x - 2y = 13
Solve the system by substitution. The first equation is already solved for y. Substitute the
2
expression __
​  14 ​ ​​(x - 3)​​ ​ for y in the second equation.
2
B
(
)
2
3x - 2 ​ _
​  1 ​​​ (x - 3)​​ ​ ​ = 13
4
Now, solve for x.
(
)
2
13 = 3x - 2​ _
​  1 ​​​ (  x - 3)​ ​ ​ ​
4
13 = 3x -
(
(​​ x - 3)​ ​2​
)​
13 = 3x - _
​  1 ​  ​
2
​  9 ​ 
13 = 3x - _
​  1 ​  ​x 2​ ​ + 3x - _
2
2
13 = - _
​  1 ​  ​x 2​ ​ +
2
-_
​  9 ​ 
2
​  35 ​  
 0 = - _
​  1 ​  ​x 2​ ​ + 6x - _
2
2
 0 = x​  2​ ​
( )​(​ x )​
  x = (​
)​or x = (​ )​
 0 = ​x
The line and the parabola intersect at two points. Use the x-coordinates of the intersections to find
the points.
Solve 3x - 2y = 13 for y.
3x - 2y = 13
-2y = 13 - 3x
y=
13 - 3 ​
⋅ 5 
y = -​  _
 
2
13
15 
_
= -​ 
 ​
 
2
-2    ​
= -​ _
2
= 1
13 - 3 ​
⋅ 7 
y = -​  _
 
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Find y when x = 5 and when x = 7.
2
13
21 
_
= -​ 
 ​
 
2
- 8    ​
= -​ _
2
= 4
So the solutions to the system are .
Reflect
5.
How can you check algebraic solutions for reasonableness?
Module 5
258
Lesson 2
Your Turn
Solve the given linear-quadratic system algebraically.
6.
1 y2
⎧x - 6 = -__
6
⎨⎩ 2x + y = 6
Explain 3
7.
⎧x - y = 7
⎨x
⎩
2
-y= 7
Solving Real-World Problems
You can use the techniques from the previous examples to solve real-world problems.
Example 3
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Shutterstock

Solve each problem.
A tour boat travels around an island in an ellipical pattern. When the
boat is directly north or south of the island, it is 6 units away; and
when it is directly east or west of the island, it is 5 units away. Taking
the island as (0, 0), a fishing boat approaches the island on a path that
can be modeled by the equation 3x - 2y = -8. Is there a danger of
collision? If so, where?
Write the system of equations.
The ellipse with vertical major axis of length
6 and horizontal minor
y2
x2
__
axis of length 5 has the equation __
+
=
1.
Multiplying both sides
25
36
2
2
by 900, this is equivalent to 36x + 25y = 900.
⎧ 2
36x + 25y 2 = 900
⎨
3x - 2y = -8
⎩
Solve the second equation for x.
3x - 2y = -8
3x = 2y - 8
2y - 8
x = _
3
Module 5
259
Lesson 2
Substitute for x in the first equation.
36​x 2​ ​ + 25​y 2​ ​ = 900
(
)
2y - 8
 ​
36​​ ​  _
 
 ​​ ​ + 25​y 2​ ​ = 900
3
(
2
)
4​y 2​ ​- 32y + 64
 ​
36​ ​  __
  
 
​ + 25​y 2​ ​ = 900
9
4​(4​y 2​ ​ - 32y + 64)​ + 25​y 2​ ​ = 900
16​y 2​ ​ - 128y + 256 + 25​y ​2​ = 900
41​y 2​ ​ - 128y - 644 = 0
Solve using the quadratic equation.
28 ​2​ - 4(41)(-644) ​
128 ± √
​ ​1  
 ​
y = ​  ___
   
  
2(41)
___
――― 
 
128 ±​  √122,000 ​
  
 
 ​
= ​ __
82
≈ -2.70 or 5.82
Collisions can occur when y ≈ -2.70 or y ≈ 5.82.
2y - 8
To find the x-values, substitute the y-values into x = ​  _____
   
. ​
3
2​(-2.70)​  - 8
x = __
 ​
​   
 
3
2​(5.82)​ - 8
x = ​  _
 ​
 
 
-5.40 ​
- 8 
= ​  _
 
11.64  ​
- 8 
= ​  _
 
= _
​  -13.40
 
 
 ​
= _
 ​
​  3.64
 
≈ -4.47
≈ 1.21
3
3
3
3
© Houghton Mifflin Harcourt Publishing Company
3
So the boats could collide at approximately (​ -4.47, -2.70)​or (​ 1.21, 5.82)​.
Module 5
260
Lesson 2
B
The signal from a radio station can be detected up to 45 units of distance away. Taking
the location of the station as (0, 0), a stretch of highway near the station is modeled by the
1
equation y - 15 = __
​  20
   ​x. At which points, if any, does a car on the highway enter and exit the
broadcast range of the station?
Write the system of equations.
With the station at (0, 0), the signal will reach out in a circle. The equation of a circle with
radius 45 is x​  ​2​+ ​y 2​ ​= ​45 ​2​= 2025.
⎧  ​x ​2​ + ​y 2​ ​= 2025 ​ ​​        
  
 ​
1  ​ x
  y - 15 = _
​ 
⎩
⎨ 
20
Solve the second equation for y.
y - 15 = _
​  1  ​ x
20
y =
Substitute for x in the first equation.
​x ​2​+ ​y 2​ ​ = 2025
(
)​​​ = 2025
2
​x  2​ ​ + ​​
​x  2​ ​ +
= 2025
​x  2​ ​ + _
​  3 ​x  + 225 = 2025
2
3
401 2 _
_
  ​​x ​ ​ + ​    ​x  400
2
=0
© Houghton Mifflin Harcourt Publishing Company
401​x 2​ ​ + 600x - 720000 = 0
―――――――――
   
Solve using the quadratic formula.
-600 ± ​  √ 6​00 ​2​ - 4​(401)​​(-720000)​ ​
 ​
y = ​  ____
   
  
2​(401)​
≈
or
(rounded to the nearest hundredth)
To find the y-values, substitute the x-values into y = _
​  1  ​ x + 15.
20
The car will be within the radio station’s broadcast area between .
Module 5
261
Lesson 2
Your Turn
8.
1
An asteroid is traveling toward Earth on a path that can be modeled by the equation y = __
​  28
  ​x
  - 7. It
y​  2​ ​
​x ​ ​
  ​ + __
​  51 ​ = 1. What are the
approaches a satellite in orbit on a path that can be modeled by the equation __
​ 49
approximate coordinates of the points where the satellite and asteroid might collide?
2
9.
The owners of a circus are planning a new act. They want to have a trapeze artist catch another acrobat
in mid-air as the second performer comes into the main tent on a zip-line. If the path of the trapeze can
be modeled by the parabola y = __
​  14 ​​x 2​ ​ + 16 and the path of the zip-line can be modeled by y = 2x + 12, at
what point can the trapeze artist grab the second acrobat?
10. A parabola opens to the left. Identify an infinite set of parallel lines that each of which intersect the parabola
only once.
 |
 ⎧
 ⎫
11. If a parabola can intersect the set of lines ⎨ 
​​  ​   in 0, 1, or 2 points, what do you know about the
 ⎩​ ​x   = a​|  |​a  ​∈R  ​ ⎬ 
⎭
parabola?
12. Essential Question Check-In How can you solve a system composed of a linear equation in two
variables and a quadratic equation in two variables?
Module 5
262
Lesson 2
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Elaborate Evaluate: Homework and Practice
1.
How many points of intersection
are on the graph?
8
2.
y
How many points
of intersection are there
on the graph
= ​x 2​ ​ + 3x - 2
 ⎧y      
 ​?
of ⎨ 
​   ​   
⎩y - x = 4
4
x
-8
-4
0
4
8
-4
Solve each given linear-quadratic system graphically. If necessary, round to the nearest integer.
3.
⎧  y = -​​(x - 2)2​​ ​ + 4
​  ​​         
  
 ​
⎨ 
⎩ y = -5
8
 ⎧y - 3 = (​​ x - 1)​ ​ ​
      
 ​
⎨ 
 ​ ​​    
⎩ 2x + y = 5
2
4.
y
8
4
y
4
x
-8
-4
0
4
x
-8
8
-4
-4
© Houghton Mifflin Harcourt Publishing Company
-8
5.
4
8
-4
-8
 ⎧x = ​y 2​ ​ - 5
     
 ​
⎨ 
 ​ ​​    
⎩ -x + 2y = 12
8
0
 ⎧ x - 4 = (​​ y + 1)​ ​ ​
      
 ​
⎨ 
 ​ ​​   
⎩ 3x - y = 17
2
6.
y
8
y
4
4
x
x
-8
-4
0
4
-8
8
-4
4
8
-4
-8
-8
Module 5
-4
0
263
Lesson 2
⎧  ​​(y - 4)​ ​ ​ + ​x 2​ ​ = -12x - 20
 ​ 8.
   
 ​⎨ ​           
⎩x = y
2
7.
8
⎧  5 - y = ​x 2​ ​ + x
​ ​       
 ​
  
 ⎨ 
⎩y + 1 = _
​  3 ​ x
4
y
8
4
4
x
-8
-4
0
4
y
x
8
-8
-4
-8
-4
0
4
-4
-8
⎧  y - 5 = (​​ x - 2)​ ​2​
10. ​ ⎨ ​​        
  
 ​
⎩ x + 2y = 6
⎧  ​y 2​ ​ - 26 = -​x 2​ ​
  
11. ​⎨ 
 ​
  ​  x     
⎩ -y=6
⎧  y - 3 = x​  2​ ​ -2x
      
 ​
  
12. ⎨ 
 ​ ​  2x
⎩ +y=1
⎧  y = ​x 2​ ​ + 1
13.  ⎨ 
​ ​       ​ 
⎩y - 1 = x
⎧  y = ​x 2​ ​ + 2x + 7
14. ​ ⎨ ​        
  
 ​
⎩y - 7 = x
264
© Houghton Mifflin Harcourt Publishing Company
Solve each linear-quadratic system algebraically.
⎧  6x + y = -16
 ​
9. ⎨ 
  
 ​ ​​       
2
⎩ y + 7 = x​  ​ ​
Module 5
8
Lesson 2
Write and solve a system of equations to find the solutions.
15. Jason is driving his car on a highway at a constant rate of 60 miles per hour when he
passes his friend Alan whose car is parked on the side of the road. Alan has been
waiting for Jason to pass so that he can follow him to a nearby campground. To catch
up to Jason's passing car, Alan accelerates at a constant rate. The distance d, in miles,
that Alan's car travels as a function of time t, in hours, since Jason's car has passed is
given by d = 3600​t ​2​. How long does it takes Alan's car to catch up with Jason's car?
16. The flight of a cannonball toward a hill is described by the
parabola y = 2 + 0.12x - ​0.002x ​2​.
The hill slopes upward, at a grade of 15% (meaning it rises
15 feet for every 100 horizontal feet it runs).
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Where on the hill does the cannonball land?
17. A ball is launched into the air from the ground with an initial vertical velocity of
64 ft/sec. At the same time a balloon is released from a height of 40 feet, and it rises at
the rate of 8 ft/sec. Write and solve a system of equations to determine when the ball
and the balloon are at the same height.
Module 5
265
Lesson 2
18. The range of an ambulance service is twenty miles in any
direction from its station. Taking the ambulance's station
to be at (0, 0), a straight road within the service area is
represented by y = 3x + 20. Find the length in miles of
the road that lies within the range of the ambulance service
(round your answer to the nearest hundredth).
Recall ____
that the distance formula is
​x 2​ ​ - ​x 1​ ​)2​​ ​ + (​​ ​y 2​ ​ - ​y 1​ ​)2​​ ​ ​.
d=√
​ ​​(  
 ⎧y = (​​ x - 2)​​ ​
⎨ 
​
   
 ​ ​​ y     
⎩ = -5x - 8
2
B. ​
(0, -2)​ (​ 5, 3)​
 ⎧4y = 3x
⎨ 
   
​
C. ​
(2, 0)​
 ​ ​​      
2
2
⎩ ​x ​ ​ + ​y ​ ​ = 25
⎧  y = (​​ x - 2)​ ​2​
​⎨ 
 ​ 
D. No solution
  ​​      
⎩y = 0
⎧  y - 7 = x​  2​ ​ - 5x
  
 ​graphically and
20. A student solved the system ⎨ 
 ​ ​​ y      
⎩ - 2x = 1
determined the only solution to be (1, 3). Was this a reasonable
8
y
4
answer? How do you know?
x
-8
-4
4
8
-8
Module 5
266
Lesson 2
© Houghton Mifflin Harcourt Publishing Company · Image Credits: ©Glen
Jones/Shutterstock
19. Match the equations with their solutions.
⎧  y = x - 2
  
 
⎨ 
​ A. ​
(4, 3)​ (​ -4, -3)​
 ​ ​​       
2
⎩ -​x ​ ​ + y = 4x - 2
H.O.T. Focus on Higher Order Thinking
21. Explain the Error A student was asked to come up with a system of equations, one
⎧   ​y 2​ ​ = -​​(x + 1)2​​ ​ + 9
linear and one quadratic, that has two solutions. The student gave ⎨ 
  
 ​
 ​ ​​        
2
⎩ y = ​x ​ ​ - 4x + 3
as the answer. What did the student do wrong?
22. Analyze Relationships The graph shows a quadratic function and a
linear function y = d. If the linear function were changed to y = d + 3,
how many solutions would the new system have? If the linear function
were changed to y = d - 5, how many solutions would the new system
have? Give reasons for your answers.
8
y
4
x
-8
-4
0
4
8
-4
-8
23. Make a Conjecture Given y = ​100x 2​ ​ and y = 0​ .0001x ​2​, what can you say about any
line that goes through the vertex of each but is not horizontal or vertical?
© Houghton Mifflin Harcourt Publishing Company
24. Communicate Mathematical Ideas Explain why a system of a linear equation
and a quadratic equation cannot have an infinite number of solutions.
Module 5
267
Lesson 2
Lesson Performance Task
Suppose an aerial freestyle skier goes off a ramp with her path represented by the equation
2
y = -0.024​​( x - 25 )​ ​ ​ + 40. If the surface of the mountain is represented by the linear equation
y = -0.5x + 25, find the distance in feet the skier lands from the beginning of the ramp.
© Houghton Mifflin Harcourt Publishing Company · Image Credits:
©EpicStockMedia/Alamy
Module 5
268
Lesson 2