1. Solve a quadratic system by elimination
EXAMPLE 3
Solve a quadratic system by elimination
Solve the system by elimination.
9x2 + y2 – 90x = 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 = 0
x2 – y – 16 = 0
10x – 90x = 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

2. EXAMPLE 3
Solve a quadratic system by elimination
When x = 4, y = 0. When x = 5, y = ±3.
ANSWER
The solutions are (4, 0), (5, 3), and (5, 23), as shown.

3. EXAMPLE 4
Solve a real-life quadratic system
Navigation
A ship uses LORAN (long-distance radio navigation) to find its position.Radio signals from stations A and B locate the ship on the blue hyperbola, and signals from stations B and C locate the ship on the red hyperbola.
The equations of the hyperbolas are given below. Find the ship’s position if it is east of the y - axis.

4. Solve a real-life quadratic system
EXAMPLE 4
Solve a real-life quadratic system
x2 – y2 – 16x + 32 = 0
Equation 1
– x2 + y2 – 8y + 8 = 0
Equation 2
SOLUTION
STEP 1
Add the equations to eliminate the x2 - and y2 - terms.
x2 – y2 – 16x = 0
– x2 + y2
– 8y + 8 = 0
– 16x – 8y + 40 = 0
Add.
y = – 2x + 5
Solve for y.

5. Solve a real-life quadratic system
EXAMPLE 4
Solve a real-life quadratic system
STEP 2
Substitute – 2x + 5 for y in Equation 1 and solve for x.
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.
x = – 1 or x = 73
Zero product property

6. ( ). ( ). EXAMPLE 4 Solve a real-life quadratic system STEP 3
Substitute for x in y = – 2x + 5 to find the solutions
(–1, 7) and , 73 13 ( ).
ANSWER
Because the ship is east of the y - axis, it is at , 73 13 ( ).

7. GUIDED PRACTICE for Examples 3 and 4 Solve the system. 7.
–2y2 + x + 2 = 0
x2 + y2 – 1 = 0
SOLUTION
–2y2 + x + 2 = 0
x2 + y2 – 1 = 0
Multiply 2nd equation by 2 to eliminate y2 term and obtain quadratic equation.
–2y x + 2 = 0
2y2 + 2x – 2 = 0
2x2 + x = 0
Add.

8. GUIDED PRACTICE for Examples 3 and 4 x(2x + 1) = 0 -1 x = 0 or x = 2
Factor
x = 0 or x = -1 2
Zero product property 2 3.
When x = 0, y = ± 1. When x = , y = ± -1

9. GUIDED PRACTICE
for Examples 3 and 4
Solve the system. 8.
x2 + y2 – 16x + 39 = 0
x2 – y2 – 9 = 0
SOLUTION
x2 + y2 – 16x + 39 = 0
x2 – y2 – = 0
2x2 – 16x – 30 = 0

10. GUIDED PRACTICE for Examples 3 and 4 2x2 – 16x – 30 = 0
Factor
x = 3 or x = 5
Zero product property
When x = 3, y = 0. When x = 5, y = ± 4

11. GUIDED PRACTICE for Examples 3 and 4 Solve the system. 9.
x2 + 4y2 + 4x + 8y = 8
y2 – x + 2y = 5
SOLUTION
x2 + 4y2 + 4x + 8y = 8
4y2 + 4x  8y =  20
Multiply 2nd equation by 4 to obtain a quadratic equation.
x2 + 8x = –12
Add.
(x + 2) (x + 6) = 0
Factor
x = – 2 or x = – 6
Zero product property

12. GUIDED PRACTICE
for Examples 3 and 4
When x = –2, y = –3. When x = – 6, y = – 1
ANSWER
The solutions are (– 6, 1), (– 2, –3), and (–2, 1).

13. Add the equations to eliminate the x2 - and y2 - terms.
GUIDED PRACTICE
for Examples 3 and 4
10.
WHAT IF? In Example 4, suppose that a ship’s LORAN system locates the ship on the two hyperbolas whose equations are given below. Find the ship’s location if it is south of the x-axis.
x2 – y2 – 12x + 18 = 0
Equation 1
y2 – x2 – 4y + 2 = 0
Equation 2
SOLUTION
STEP 1
Add the equations to eliminate the x2 - and y2 - terms.
x2 – y2 – 12x = 0
– x2 + y2
– 4y + 2 = 0
–12x – 4y + 20 = 0
Add.

14. Substitute –3x + 5 for y in equation 1 and solved for x.
GUIDED PRACTICE
for Examples 3 and 4
y = – 3x + 5
Solve for y.
STEP 2
Substitute –3x + 5 for y in equation 1 and solved for x.
x2 – y2 – 12x + 18 = 0
Equation 1
x2 – (– 3x + 5)2 – 12x + 18 = 0
Substitute for y.
8x2 –18x + 7 = 0
Simplify.
(2x – 1)(4x – 7) = 0
Factor.
x = or x = 74 1 2
Zero product property

15. ) ( ). ). ( EXAMPLE 4 Solve a real-life quadratic system STEP 3
Substitute for x in y = – 3x + 5 to find the solutions
and, 7 2 1 , ) ( 74 –1 ). 4 ).
ANSWER
Because the ship is south of the x - axis, it is at , 74 ( –1 4