Nonlinear Systems of Equations Objective: To solve a nonlinear system of equations.

Review – What? System – 2 or more equations together Solution of system – any ordered pair that makes all equations true Possible solutions: One point More than one point No solution Infinite solutions

Review - How? What methods have we used to solve linear systems of equations? Graphing Substitution Elimination

Review Steps for using SUBSTITUTION Solve one equation for one variable. (Hint: Look for an equation already solved for a variable or for a variable with a coefficient of 1 or -1.) Substitute into the other equation. Solve this equation to find a value for the variable. Substitute again to find the value of the other variable. Check.

Review Solve using Substitution. 2x – y = 6 y = 5x

Review STEPS for ELIMINATION

Review Solve using elimination. 3x + 5y = 11 2x + 3y = 7

What’s New? A non-linear system is one in which one or more of the equations has a graph that is not a line. With non-linear systems, the solution could be one or more points of intersection or no point of intersection. We’ll solve non-linear systems using substitution or elimination. A graph of the system will show the points of intersection.

Solve the following system of equations: An Example… Solve the following system of equations:

An Example… We use the substitution method. First, we solve equation (2) for y.

An Example… Next, we substitute y = 2x  3 in equation (1) and solve for x:

An Example… … x = 0 x = 12 / 5 y = 2x  3 y = 2(0)  3 y = 3 Now, we substitute these numbers for x in equation (2) and solve for y. x = 0 x = 12 / 5 y = 2x  3 y = 2(0)  3 y = 3 SOLUTIONS (0, 3) and

An Example… Visualizing the Solution Check: (0, 3) Check:

See Example 1, page 747 Check Point 1: Solve by substitution. x2 = y – 1 4x – y = -1

Example 2 Check Point 2: Solve by substitution. x + 2y = 0

Another example to watch… Solve the following system of equations: xy = 4 3x + 2y = 10

Solve xy = 4 for y. Substitute into 3x + 2y = 10.

Use the quadratic formula (or factor) to solve:

Substitute values of x to find y. 3x + 2y = 10 x = 4/3 x = 2 The solutions are (4/3, 3) and (2, 2). Visualizing the Solution

Need to watch another one? Solve the system of equations:

Solve by elimination. Multiply equation (1) by 2 and add to eliminate the y2 term.

Substituting x = 1 in equation (2) gives us: x = 1 x = -1 The possible solutions are (1, 3), (1, 3), (1, 3) and (1, 3).

Visualizing the Solution All four pairs check, so they are the solutions.

Example 3 Check Point 3: Solve by elimination. 3x2 + 2y2 = 35

Example 4 Check Point 4: Solve by elimination. y = x2 + 5 x2 + y2 = 25