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Quadratic System Elimination: Solve Equations and Find Ship's Position

Learn how to solve a quadratic system by elimination and find the position of a ship using real-life examples. Solve equations and eliminate terms to obtain a quadratic equation in x and y.

stephenmanuel
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Quadratic System Elimination: Solve Equations and Find Ship's Position

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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, –3),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. -1 3 2 2 for Examples 3 and 4 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

  8. for Examples 3 and 4 GUIDED PRACTICE Solve the system. 9. x2 + 4y2 + 4x + 8y = 8 y2 – x + 2y = 5 ANSWER (–6, –1), (–2, –3),(–2, 1).

  9. 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. 74 ( –1 4 for Examples 3 and 4 GUIDED PRACTICE x2 – y2 – 12x + 18 = 0 Equation 1 y2 – x2 – 4y + 2 = 0 Equation 2 ) , ANSWER

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