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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 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.
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
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).
Holt McDougal Algebra 1
Nonlinear Systems
Remember!
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.
Holt McDougal Algebra 1
Nonlinear Systems
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
Holt McDougal Algebra 1
Substitute -3x = 3 for y in the first
equation
Nonlinear Systems
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
Holt McDougal Algebra 1
Subtract -3x + 4 from both
sides
Nonlinear Systems
Example 3 : Solving a Nonlinear System
A
3x - y = 1
y = x2 + 4x - 7
Holt McDougal Algebra 1
Nonlinear Systems
Check It Out! Example 3
1. Solve each system by elimination. Check your answers..
a
2x - y = 2
y = x2 - 5
Write the system to align the y-terms
2x - y = 2
y = x2 - 5
2x = x2 - 3
-2x
-2x
Holt McDougal Algebra 1
Add to eliminate y
Subtract 2x from booth sides
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
Helpful Hint
When t = 0, the ball and elevator are at the same height
because they are both at ground level.
Holt McDougal Algebra 1
Nonlinear Systems
Examples
Solve each system.
1.
2.
Holt McDougal Algebra 1
y = x2 - 4x + 3
y=x-1
y = 2x2 - 9x - 5
y = -3x + 3
(1, 0), (4, 3)
(-1, 6), (4, -9)
Nonlinear Systems
Examples
3.
y = x2 + 2x - 3
x-y=5
no solution
4.
y = x2 - 7x + 10
2x - y = 8
(3, -2), (6, 4)
Holt McDougal Algebra 1
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