Warm Up

1
Solve Quadratic Systems
Warm Up
Lesson Presentation
Lesson Quiz
2
Warm-Up
Solve the system by substitution.
(2, 3)
ANSWER
(2, 1)
ANSWER
ANSWER
120 students and 80 nonstudents
3
Example 1
Solve the system using a graphing calculator.
y2 7x 3 0
Equation 1
2x y 3
Equation 2
SOLUTION
y2 7x 3 0
2x y 3
y2 7x 3
y 2x 3
y 2x 3
Equation 1
Equation 2
4
Example 1
5
Guided Practice
Solve the system using a graphing calculator.
(3,2) , (2, 3)
ANSWER
no solutions
ANSWER
(1.57, 3.23) , (22.9, 11.8).
ANSWER
6
Example 2
Solve the system using substitution.
x2 y2 10
Equation 1
y 3x 10
Equation 2
SOLUTION
Substitute 3x 10 for y in Equation 1 and solve
for x.
Equation 1
x2 y2 10
x2 (3x 10)2 10
Substitute for y.
x2 9x2 60x 100 10
Expand the power.
10x2 60x 90 0
Combine like terms.
x2 6x 9 0
Divide each side by 10.
(x 3)2 0
Perfect square trinomial
x 3
Zero product property
7
Example 2
To find the y-coordinate of the solution,
substitute x 3 in Equation 2.
y 3(3) 10 1
CHECK You can check the solution by graphing the
equations in the system. You can see from the
graph shown that the line and the circle
intersect only at the point (3, 1).
8
Guided Practice
Solve the system using substitution.
no solution.
ANSWER
(3, 4) , (3, 2)
ANSWER
(2, 0) , ( , )
ANSWER
9
Example 3
Solve the system by elimination.
9x2 y2 90x 216 0
Equation 1
x2 y2 16 0
Equation 2
SOLUTION
Add the equations to eliminate the y2 - term and
obtain a quadratic equation in x.
9x2 y2 90x 216 0
10x2 90x 200 0
Add.
x2 9x 20 0
Divide each side by 10.
(x 4)(x 5) 0
Factor
x 4 or x 5
Zero product property
10
Example 3
When x 4, y 0. When x 5, y 3.
11
Guided Practice
Solve the system.
7.
2y2 x 2 0
x2 y2 1 0
(0, 1) , ( , )
ANSWER
8.
x2 y2 16x 39 0
x2 y2 9 0
(3, 0) , (5, 4)
ANSWER

12
Guided Practice
Solve the system.
9.
x2 4y2 4x 8y 8
y2 x 2y 5
ANSWER
(6, 1), (2, 3), (2, 1).
13
Example 4
Navigation
14
Example 4
x2 y2 16x 32 0
Equation 1
x2 y2 8y 8 0
Equation 2
SOLUTION
x2 y2 16x 32 0
16x 8y 40 0
Add.
y 2x 5
Solve for y.
15
Example 4
x2 y2 16x 32 0
Equation 1
x2 (?2x 5)2 16x 32 0
Substitute for y.
3x2 4x 7 0
Simplify.
(x 1)(3x 7) 0
Factor.
Zero product property
16
Guided Practice
x2 y2 12x 18 0
Equation 1
y2 x2 4y 2 0
Equation 2
)
,
ANSWER
17
Lesson Quiz
18
Lesson Quiz
3. A system for tracking ships has just
located a ship on a path modeled by 2x2 y2 1.
Earlier the ship was located on a path modeled by
2y2 x2 1. If the ship is in the first quadrant
of a coordinate system, what is the ship's
location in the coordinate system now?