Holt McDougal Algebra 1 Nonlinear Systems Solve the following quadratic Inequality 1. 5-2x 2 ≥ -3x...
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Transcript of Holt McDougal Algebra 1 Nonlinear Systems Solve the following quadratic Inequality 1. 5-2x 2 ≥ -3x...
Holt McDougal Algebra 1
Nonlinear Systems
Solve the following quadratic Inequality
1.5-2x2 ≥ -3x
2.4x2 < 9
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)