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EXAMPLE 3

Add the equations to eliminate the y 2 - term and obtain a quadratic equation in x. x 2 – y 2 – 16 = 0. EXAMPLE 3. Solve a quadratic system by elimination. Solve the system by elimination. 9 x 2 + y 2 – 90 x + 216 = 0. Equation 1. x 2 – y 2 – 16 = 0. Equation 2.

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EXAMPLE 3

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  1. Add the equations to eliminate the y2- term and obtain a quadratic equation in x. x2 – y2– 16 = 0 EXAMPLE 3 Solve a quadratic system by elimination Solve the system by elimination. 9x2 + y2 – 90x + 216 = 0 Equation 1 x2 – y2 – 16 = 0 Equation 2 SOLUTION 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 orx = 5 Zero product property

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

  3. 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. EXAMPLE 4 Solve a real-life quadratic system Navigation

  4. STEP 1 Add the equations to eliminate the x2 - and y2 - terms. – x2 + y2 – 8y + 8 = 0 EXAMPLE 4 Solve a real-life quadratic system 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.

  5. STEP 2 Substitute – 2x + 5for yin Equation 1 and solve for x. x = – 1 orx = 73 EXAMPLE 4 Solve a real-life quadratic system 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

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

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

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

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

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

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

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

  13. 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. STEP 1 Add the equations to eliminate the x2 - and y2 - terms. – x2 + y2 – 4y + 2 = 0 x2 – y2 – 12x + 18 = 0 for Examples 3 and 4 GUIDED PRACTICE x2 – y2 – 12x + 18 = 0 Equation 1 y2 – x2 – 4y + 2 = 0 Equation 2 SOLUTION –12x – 4y + 20 = 0 Add.

  14. STEP 2 Substitute –3x + 5 for y in equation 1 and solved for x. 74 1 x = orx = 2 for Examples 3 and 4 GUIDED PRACTICE y = – 3x + 5 Solve for y. 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. Zero product property

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

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