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Solving Systems of Nonlinear Equations - Precalculus

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Terms in this set (22)

  • What is the substitution method for solving nonlinear systems?
    Replace one variable with an expression from one equation into the other, then solve the resulting equation.
  • How do you solve the system: x - y = 3 and (x - 2)^2 + (y + 3)^2 = 4 using substitution?
    Express x = y + 3, substitute into the circle equation, expand and solve the quadratic for y, then find x.
  • What is the addition method for solving nonlinear systems?
    Add or subtract equations to eliminate one variable, then solve for the remaining variable.
  • How to solve the system 4x^2 + y^2 = 13 and x^2 + y^2 = 10 using addition?
    Subtract the second from the first to get 3x^2 = 3, solve for x, then substitute back to find y.
  • How to solve the system x + y = 4 and y = x^2 - 2 using substitution?
    Substitute y in the first equation: x + (x^2 - 2) = 4, simplify to x^2 + x - 6 = 0, solve for x, then find y.
  • What are the solutions to x^2 + x - 6 = 0?
    x = -3 and x = 2.
  • How to find y when x = -3 or 2 in y = x^2 - 2?
    For x = -3, y = 9 - 2 = 7; for x = 2, y = 4 - 2 = 2.
  • How to solve the system -10x - y = 32 and y = x^2 - 8 using substitution?
    Substitute y in the first: -10x - (x^2 - 8) = 32, simplify to x^2 + 10x + 24 = 0, solve for x, then find y.
  • What are the solutions to x^2 + 10x + 24 = 0?
    x = -6 and x = -4.
  • How to solve the system 4x^2 + y^2 = 16 and 4x^2 - y^2 = 16 using addition?
    Add the equations: 8x^2 = 32, solve for x, then substitute back to find y.
  • What are the solutions for x and y in the system 4x^2 + y^2 = 16 and 4x^2 - y^2 = 16?
    x = ±2 and y = 0.
  • How to solve the system x + y = 11 and y = x^2 - 10x + 25 using substitution?
    Substitute y in the first: x + (x^2 - 10x + 25) = 11, simplify and solve quadratic for x, then find y.
  • What are the solutions to x^2 - 9x + 14 = 0?
    x = 2 and x = 7.
  • What are the solution pairs for the system x + y = 11 and y = x^2 - 10x + 25?
    (2, 9) and (7, 4).
  • How to solve the system x^2 + y^2 = 145 and x + y = 17 using substitution?
    Express y = 17 - x, substitute into the circle equation, simplify and solve quadratic for x, then find y.
  • What are the solution pairs for the system x^2 + y^2 = 145 and x + y = 17?
    (8, 9) and (9, 8).
  • How to solve the system x^2 + y^2 = 1 and x^2 - y^2 = 1 using addition?
    Add equations: 2x^2 = 2, solve for x, then find y.
  • What are the solution pairs for the system x^2 + y^2 = 1 and x^2 - y^2 = 1?
    (1, 0) and (-1, 0).
  • How to solve the system x^2 + y^2 = 16 and 16x^2 + 9y^2 = 144 using substitution?
    Express y^2 = 16 - x^2, substitute into second equation, solve for x, then find y.
  • What are the solution pairs for the system x^2 + y^2 = 16 and 16x^2 + 9y^2 = 144?
    (0, 4) and (0, -4).
  • How to solve the system x^2 - y^2 = 16 and 25x^2 + 16y^2 = 400 using substitution?
    Express y^2 = x^2 - 16, substitute into second equation, solve for x, then find y.
  • What are the solution pairs for the system x^2 - y^2 = 16 and 25x^2 + 16y^2 = 400?
    (4, 0) and (-4, 0).