Holt McDougal Algebra 1

Nonlinear Systems

Solve the following quadratic Inequality

1.5-2x2 ≥ -3x

2.4x2 < 9

Holt McDougal Algebra 1

Nonlinear Systems

Section 11: Nonlinear systems

Holt McDougal Algebra 1

Nonlinear Systems

What methods can you use to solve a system that includes a linear equation and a quadratic equation?

Holt McDougal Algebra 1

Nonlinear Systems

Standards in this sectionText book pages: P548-555

MCC9-12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Holt McDougal Algebra 1

Nonlinear Systems

Nonlinear system of equations- a system in which at least one of the equations is non linear.

Holt McDougal Algebra 1

Nonlinear Systems

A system made up of a linear equation and a quadratic equation can have no solution, one solution, or two solutions, as shown below.

Holt McDougal Algebra 1

Nonlinear Systems

Example 1: Solving a Nonlinear System by Graphing

y = x2 + 4x + 3y = x + 3

Solve the system by graphing. Check your answer.

Step 1 Graph y = x2 + 4x + 3.The axis of symmetry is x = –2. The vertex is (–2, –1).The y-intercept is 3.Another point is (–1, 0).

Holt McDougal Algebra 1

Nonlinear Systems

The substitution method is a good choice when either equation is solved for a variable, both equations are solved for the same variable, or a variable in either equation has a coefficient of 1 or -1.

Remember!

Holt McDougal Algebra 1

Nonlinear Systems

Example 2: Solving a Nonlinear system by substitution.

y = x2 - x - 5y = -3x + 3

Solve the system by substitution.

Both equations are solved for y, so substitute one expression for y into the other equation for y.

-3x + 3 = x2 –x -5 Substitute -3x = 3 for y in the first equation

Holt McDougal Algebra 1

Nonlinear Systems

1. Solve the system by substitution. Check your answer.

Check It Out! Example 2

y = 3x2 - 3x + 1y = -3x + 4

Both equations are solved for y, so substitute one expression for y into the other equation for y.

-3x + 4 = 3x2 - 3x + 1 Subtract -3x + 4 for y in first equation.

0 = 3x2 - 3 Subtract -3x + 4 from both sides

Holt McDougal Algebra 1

Nonlinear Systems

Example 3 : Solving a Nonlinear System

3x - y = 1y = x2 + 4x - 7

A

Holt McDougal Algebra 1

Nonlinear Systems

1. Solve each system by elimination. Check your answers..

Check It Out! Example 3

2x - y = 2y = x2 - 5

a

Write the system to align the y-terms

2x - y = 2 y = x2 - 5

2x = x2 - 3

Add to eliminate y

-2x -2x Subtract 2x from booth sides

Holt McDougal Algebra 1

Nonlinear Systems

Example 4: Physics Application

The increasing enrollment at South Ridge High School can be modeled by the equation E(t) = -t2 + 25t + 600, where t represents the number of years after 2010. The increasing enrollment at Alta Vista High School can be modeled by the equation E(t) = 24t + 570. In what year will the enrollments at the two schools be equal?

Holt McDougal Algebra 1

Nonlinear Systems

When t = 0, the ball and elevator are at the same height because they are both at ground level.

Helpful Hint

Holt McDougal Algebra 1

Nonlinear Systems

Examples

(1, 0), (4, 3)

Solve each system.

y = x2 - 4x + 3y = x - 1

1.

2. y = 2x2 - 9x - 5y = -3x + 3

(-1, 6), (4, -9)

Holt McDougal Algebra 1

Nonlinear Systems

Examples

no solutiony = x2 + 2x - 3x - y = 5

3.

4. y = x2 - 7x + 102x - y = 8

(3, -2), (6, 4)

Homework

Text book: (Exercises 16-4) P 552 # 1-9

Worksheets: Nonlinear systems practice I, II, and III

Coach book: p 236-237 #1-9