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3.9 – Nonlinear Systems of Equations


  • Define nonlinear system of equations.
  • Determine the number of solutions that a system of nonlinear equations has when given a graph of the system.
  • List the steps for solving systems of nonlinear equations using the substitution method.
  • Solve systems of nonlinear equations.
  • Use algebra to isolate a variable in a nonlinear equation.


Key Terms

  • Quadratic Formula – A formula used to solve quadratic equations.
    • The solutions to the quadratic equation ax^2+bx+c=0 are given by the formula x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}.
  • Isolate – To separate from others.
    • To get the variable alone on one side of the equation or inequality.
  • Nonlinear System of Equations – A group of nonlinear equations that have the same variables and are used together to solve a problem.
    • The equations may be some combination of linear equations and nonlinear equations (all may be nonlinear, or only one may be nonlinear).
  • System of Linear Equations – A group of linear equations that have the same variables and are used together to solve a problem.
    • A linear equation can be written in the form y = mx + b.
    • The graph of a linear equation is a straight line.



Solutions of Nonlinear Systems of Equations
  • The first step in solving a system of nonlinear equations by substitution is often to isolate a variable in one of the equations.
  • To solve a system of equations in which there is no linear equation to start with, you can sometimes begin by isolating and substituting a variable that is squared in both equations.
  • Some nonlinear equations represent real-world situations.
    • In the real world, many x-values will start at zero and go up.
    • So, if you get a negative x-value as a solution for a frog jumping forward, a human being shot out of a cannon, or you throwing a ball forward, there won’t be any negative x-values.
  • Graphs for nonlinear systems of equations:

Alg2A 03.09 - Visual Solutions

  • Example 1:  To solve the system of equations below, Kira isolated the variable y in the first equation and then substituted it into the second equation. What was the resulting equation?


Step 1: Isolate y in the 1st equation (divide by 3 on both sides).


Step 2: Substitute the y-value of “4x” into the 2nd equation.


Step 3: Square the 4x term.


  • Example 2: How many solutions are there to the following system of equations?

Alg2A 03.09 - HowManySolutions

Answer: 4

Look for where the ellipse and hyperbola intersect.

  • Example 3: Isolate a variable term

Alg2A 03.09 - IsolteX2

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