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CC-18
Solving LinearQuadratic Systems
Objective
Content Standards
A.REI.7 Solve a simple system consisting of a linear
equation and a quadratic equation in two variables . . .
A.REI.11 Explain why the x-coordinates of the points
where the graphs of the equations y f(x) and
y g(x) intersect are the solutions of the equation
f(x) g(x) . . .
To solve systems of linear and quadratic equations
MATHEMATICAL
PRACTICES
Essential Understanding A linear-quadratic system is made up of a linear
equation and a quadratic equation. You can solve this type of system algebraically or
graphically. A linear-quadratic system can have zero, one, or two solutions.
The line and parabola do
not intersect, so the linearquadratic system has no
solution.
The line and parabola have
one point of intersection,
so the linear-quadratic
system has one solution.
The line and parabola have
two intersection points, so
the linear-quadratic system
has two solutions.
CC-18
Solving Linear-Quadratic Systems
1
Problem 1 Solving a Linear-Quadratic System
How is solving this
system like solving a
system of two linear
equations?
In both cases, you need
to find each point where
the graphs of the two
equations intersect.
What are the solutions of the system?
W
y 2x 2
8
y x2 x 6
4
Method 1: Solve by graphing.
Step 1 Graph both equations on the same coordinate plane.
x
8 4
Step 2 Identify the point(s) of intersection, if any. The two
points of intersection are (4, 6) and (=1, =4).
When solving the system of equations graphically, you can identify the solutions
by locating each point of intersection of the two graphs. If you solve the system
algebraically, the ordered pair solutions you obtain represent those points of
intersection.
Method 2: Solve algebraically.
Step 1 Write a single equation containing only one variable.
y x2 x 6
2x 2 x2 x 6
2x 2 2x x 6 2x
Substitute 2x 2 for y.
Subtract 2x from each side.
2 x2 3x 6
2 2 x2 3x 6 2
0 x2 3x 4
Add 2 to each side.
Write in standard form.
Step 2 Factor and solve for x.
0 (x 4)(x 1)
Factor.
x40
or
Zero-Product Property
x4
or
x10
x 1
Solve for x.
Step 3 Find the corresponding y-values. You may use either of the original
equations.
y 2x 2 2(4) 2 6
y 2x 2 2(1) 2 4
The solutions of the system are (4, 6) and =1, =4).
Got It? 1. What are the solutions of the system?
y x2 4x 3
y x 1
2
Common Core
O
4
8
The solutions of the system are (4, 6) and (=1, =4).
x2
y
4
8
Solutions of systems of equations do not always have integers coordinates. You can
estimate or use a graphing calculator tool to find solutions.
Problem 2
Estimating Intersection Points
What are the solutions of the system? Use a graphing calculator.
y x 2 12 x 36
3
5
y 2 x 2
Step 1 Press the y= key and enter the equations. To see
the graphs in the standard window, press zoom
6:ZStandard. You may have to press window and adust
the window settings to see all of the intersection points
for the two graphs.
Step 2 Press 2nd trace and choose 5:intersect. Move your
cursor close to one of the points of intersection. Press
enter three times to find the point of intersection.
Intersection
X523.276889 Y57.4153335
How can you check
your solutions?
You can substitute each
x-value into both original
equations. Your answers
should be close to the
estimate for the y-value
of the ordered pair.
Step 3 Repeat Step 2 to find the second intersection point.
The approximate solutions of the system are (3.3, 7.4)
and (10.2, 17.8).
Using your graphing calculator will only give estimates for the
solutions of this system. Algebraic methods for this type of problem
can give exact solutions.
Intersection
X5210.22311 Y517.834666
Got It? 2. What are the approximate solutions of the system? Use a graphing
calculator.
y x2 1
y 3x
CC-18
Solving Linear-Quadratic Systems
3
Problem 3 Using the Quadratic Formula
What are the solutions of the system? Solve by using the quadratic formula.
y 3x 7
y 2x2 7x 10
Step 1
Write a single equation containing only one variable in standard form.
3x 7 2x2 7x 10
3x 7 3x 2x2 7x 10 3x
Subtract 3x from each side.
7 2x2 4x 10
7 7 2x2 4x 10 7
Add 7 to each side.
0 2x2 4x 3
Step 2
What are the values
of a, b, and c?
a2
b4
c =3
Use the quadratic formula to solve.
x
b b2 4ac
2a
x
4 (4)2 4(2)(3)
2(2)
x
4 40
4
x
4 40
or
4
x 0.58
Step 3
Write in standard form.
Quadratic formula
Substitute the values for a, b, and c into the
quadratic formula.
Simplify.
x
4 40
4
or x =2.58
Write as two equations.
Use a calculator.
Find the corresponding y-values. You may use either of the original equations;
however, the linear equation is simpler to use.
y 3x 7
y 3(0.58) 7 or
y 3(2.58) 7
Substitute the values for x.
y 5.26
y 14.74
Simplify.
or
The approximate solutions of the system are (0.58, 5.26) and (2.58, 14.74).
Got It? 3. What are the approximate solutions of the system?
y x2 6x 17
y 4x 21
4
Common Core
Lesson Check
Do you know HOW?
Do you UNDERSTAND?
1. Use substitution to solve the system.
4. Solve the system y x2 5x 1 and y 2x 3
using the quadratic formula. What does the value
of the discriminant tell you about the number of
solutions of the system?
y x2
y 2x 3
2. Use a graphing calculator to estimate the solutions of
the system.
5. Compare and Contrast The solutions of a system
of equations solved using the quadratic formula are
y x2 2x 3
y
1
2x
+ 2 1
+ 2 3. Use the quadratic formula to solve the system. Round
to the nearest hundredth.
Practice and Problem-Solving Exercises
Practice
30
2 ,
30
2 ,
30 30 11 30
2 , and
11 30
2 ,.
When this system is solved graphically, the solutions
are (0.74, =0.12) and (=4.74, 60.12). Compare the
solutions given by each method.
y x2 4x 1
y 10x 2
A
MATHEMATICAL
PRACTICES
MATHEMATICAL
PRACTICES
See Problem 1.
Solve each system of equations.
6. y x 1
y x2 x 2
7. y 3x 1
y x 2 4x 5
8. y x 1
y x 2 2x 5
9. y x2 8x 5
y 14x 4
Solve each system using a graphing calculator. Round decimal answers to
the nearest hundreth.
10. y 0.25x 52.5
y 2x 2
11. y 16x 2 58x 23
y 12x 29
12. y 0.5x 2 1.25x 4
13. y 3x 21
16
y x 2 12x 14
y 0.75x 2
Solve each system using the quadratic formula. Round decimal answers to
the nearest hundreth.
14. y 2x 2 12x 8
y 4x 1
15. y 9x 2 91x 132
y 25x 11
16. y x 4
y 2x2 4x
17. y x 2 6
y1
CC-18
Solving Linear-Quadratic Systems
See Problem 2.
See Problem 3.
5
B
Apply
9 2
18. Which method would you use to solve the system y 10
x 15 x 11
20 and
11
y 4x 5 ? Explain.
19. Reasoning Find a linear equation with a graph that intersects the graph of
y x 2 8x 1 at exactly one point.
20. Error Analysis Marge claims that the system y 0.5x 2 0.25x 3 and y 0.5
must have two solutions because all horizontal lines will intersect a parabola in two
places. Explain why Marge’s thinking is incorrect.
21. Use a graphing calculator to solve 4x 2 x .
C
Challenge
22. The equation for a circle with the center at the origin is given by x 2 y 2 25.
Find where the line y x 1 intersects the circle.
23. Reasoning Given the quadratic function y x 2 1 and the linear function
y x b, for what values of b will the system have no solution?
Standardized Test Prep
24. If f (x) (x 4)(x 6), which equation intersects the graph of f (x) at exactly one point?
SAT/ACT
g(x) 4
g(x) 1
g(x) x 4
g(x) x 6
25. How many solutions are there for the system of equations y x 2 6x 9 and
2x y 1?
0
Short
Response
1
2
3
26. The product of two numbers is 24. If you square the smaller number and add it to the
larger number, the sum is 17. Equations modeling the two numbers are shown below.
xy 24
x 2 y 17
What is the smaller number?
Mixed Review
Write an equation in slope-intercept form for each graph shown.
27.
8
28.
y
8
4
y
4
x
8 4
O
4
8
x
8 4
O
4
4
8
8
Simplify each expression.
29. (x7y4)3
6
Common Core
30. (4n)3(3n2)2
4
8