1. Warm UP Solve the following quadratic Inequality 5-2x2 ≥ -3x

2. Unit 5: Quadratic Functions
Section 11: Nonlinear systems

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

4. Standards in this section Text 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.

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

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

7. Example 1: Solving a Nonlinear System by Graphing
Solve the system by graphing. Check your answer.
y = x2 + 4x + 3
y = x + 3
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).

8. 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!

9. Example 2: Solving a Nonlinear system by substitution.
Solve the system by substitution.
y = x2 - x - 5
y = -3x + 3
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

10. Check It Out! Example 2
1. Solve the system by substitution. Check your answer.
y = 3x2 - 3x + 1
y = -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

11. Example 3 : Solving a Nonlinear System
3x - y = 1
y = x2 + 4x - 7 A

12. Check It Out! Example 3
1. Solve each system by elimination. Check your answers..
2x - y = 2
y = x2 - 5 a
Write the system to align the y-terms
2x - y = 2
y = x2 - 5
Add to eliminate y
2x = x2 - 3
-2x
-2x
Subtract 2x from booth sides

13. 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 The increasing enrollment at Alta Vista High School can be modeled by the equation E(t) = 24t In what year will the enrollments at the two schools be equal?

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

15. Examples
Solve each system.
y = x2 - 4x + 3
y = x - 1
(1, 0), (4, 3) 1.
y = 2x2 - 9x - 5
y = -3x + 3
(-1, 6), (4, -9) 2.

16. Examples
y = x2 + 2x - 3
x - y = 5
no solution 3.
y = x2 - 7x + 10
2x - y = 8
(3, -2), (6, 4) 4.

17. Homework
Text book: (Exercises 16-4) P 552 # 1-9
Worksheets:
Nonlinear systems practice I, II, and III
Coach book: p #1-9