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Solving Nonlinear Systems of Equations

Learn how to solve systems of nonlinear equations through graphing, substitution, and elimination. Practice problems and activities provided.

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Solving Nonlinear Systems of Equations

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  1. Do Now 5/10/19 • Copy HW in your planner. • Text p. 530, #4-26 evens • Quiz sections 9.4-9.6 Friday 5/17

  2. Learning Goal Learning Target • Students will be able to solve systems of nonlinear equations by graphing, substitution, and eliminations • Students will be able to solve quadratic equations.

  3. Remember this??? “How Do You Solve a Linear System???” (1) Solve Linear Systems by Graphing (5.1) (2) Solve Linear Systems by Substitution (5.2) (3) Solve Linear Systems by ELIMINATION!!! (5.3)

  4. Remember this??? -7 -4 + (-3) -7= -7 ? ? = = 4+ 4(-3) -8 “Solve Linear Systems by Graphing” Graph to solve the system. Then check your solution algebraically. Equation 1 x + 4y = -8 Equation2 -x +y = -7 SOLUTION The lines appear to intersect at the point (4, -3). CHECK Substitute 4 forxand -3 foryin each equation. Equation 1 Equation2 Because the ordered pair (4, -3) is a solution of each equation, it is a solution of the system. x+ 4y= -8 -x+ y= -7 -8 = -8

  5. Remember this??? “Solve Linear Systems by Substituting” y = 3x + 2 Equation 1 Equation 2 x + 2y = 11 x + 2(3x + 2) = 11 x + 2y = 11 Substitute x + 6x + 4 = 11 7x + 4 = 11 x = 1 y = 3x + 2 Equation 1 Substitute value for x into the original equation y = 3(1) + 2 y = 5 (5) = 3(1) + 2 5 = 5 (1) + 2(5) = 11 11 = 11 The solution is the point (1,5). Substitute (1,5) into both equations to check.

  6. Remember this??? “Solve Linear Systems by Elimination!” Multiply First Eliminated x (-3) x – 2y = -7 -3x + 6y = 21 Equation 1 + 3x – y = 4 3x – y = 4 Equation 2 5y = 25 y = 5 x – 2y = -7 Equation 1 Substitute value for y into either of the original equations x - 2(5) = -7 x - 10 = -7 x = 3 (3) - 2(5) = -7 -7 = -7 3(3) - (5) = 4 4 = 4 The solution is the point (3,5). Substitute (3,5) into both equations to check.

  7. Section 9.6 “Solving Nonlinear Systems of Equations” The methods for solving linear equations can also be used to solve nonlinear systems of equations. When a nonlinear system consists of a linear equation and a quadratic equation, the graphs can intersect in zero, one, and two points.

  8. “Solve Nonlinear Systems by Substitution” y = -2x + 3 Equation 1 Equation 2 y = x2 + x – 1 (-2x + 3) = x2 + x – 1 y = x2 + x – 1 Substitute -2x + 3= x2 + x – 1 0 = x2 + 3x – 4 Factor 0 = (x + 4)(x – 1) x = -4 and 1 y = -2x + 3 y = -2x + 3 Equation 1 Equation 1 Substitute both values for x into the original equation y = -2(-4) + 3 y = -2(1) + 3 y = 11 y = 1 The solutions are the points (-4,11) and (1,1). Substitute both points into the equations to check.

  9. On Your OwnSolve the nonlinear system by substitution.

  10. “Solve Nonlinear Systems by Elimination!” Eliminated Eliminated y = x2 + x Equation 1 + - -y = -x – 5 y = x + 5 Equation 2 0 = x2 – 5 5 = x2 ±2.24 = x y = x2 + x y = x2 + x Equation 1 Equation 1 Substitute both values for x into the original equation y = (2.24)2 + 2.24 y = (-2.24)2 - 2.24 y ≈ 7.24 y ≈ 2.76 The solutions are the points (2.24,7.24) and (-2.24, 2.76). Substitute both points into the equations to check.

  11. On Your OwnSolve the nonlinear system by elimination.

  12. “Solve Nonlinear Systems by Graphing” y = 2x2+5x–1 Equation 1 y = x – 3 Equation2 The lines appear to intersect at the point (-1, -4). The solution is (-1, -4).

  13. On Your OwnSolve the nonlinear system by graphing.

  14. Student Journal Activity • Complete Explorations #1 & 2 on pages 301 & 302. • Then complete Extra Practice #2-12 evens on page 304 & 305.

  15. Classwork/Homework • Text p. 530, #4-26 evens

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