Chapter 6 Conic Sections Nonlinear Systems of Equations

Nonlinear Systems of Equations The graphs of the equations in a nonlinear system of equations can have no point of intersection or one or more points of intersection. The coordinates of each point of intersection represent a solution of the system of equations. When no point of intersection exists, the system of equations has no real-number solution. We can solve nonlinear systems of equations by using the substitution or elimination method.

Example Solve the following system of equations:

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

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

Example continued Now, we substitute these numbers for x in equation (2) and solve for y. x = 0 x = 12 / 5

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

Example Solve the following system of equations: xy = 4 3x + 2y = 10

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

Use the quadratic formula to solve: Example continued Use the quadratic formula to solve:

Visualizing the Solution Example continued Substitute values of x to find y. 3x + 2y = 10 x = 4/3 x = 2 The solutions are Visualizing the Solution

Example Solve the system of equations:

Example continued Solve by elimination.

Example continued Substituting x = 1 in equation (2) gives us: x = 1 x = -1 The possible solutions are

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