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Slide 8.5- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
OBJECTIVES
Nonlinear Systems of Equations and Inequalities
Learn techniques for solving nonlinear systems of equations.
Learn a procedure for solving a nonlinear system of inequalities.
SECTION 8.5
1
2
Slide 8.5- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
DefinitionsThe statements x + y > 4, 2x + 3y < 7, y ≥ x, and x + y ≤ 9 are examples of linear inequalities in the variables x and y.A solution of an inequality in two variables x and y is an ordered pair (a, b) that results in a true statement when x is replaced by a, and y is replaced by b in the inequality.The set of all solutions of an inequality is called the solution set of the inequality. The graph of an inequality in two variables is the graph of the solution set of the inequality.
Slide 8.5- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES
Step 1. Replace the inequality symbol by an
equals (=) sign.
Step 2. Sketch the graph of the corresponding
equation in Step 1. Use a dashed line
for the boundary if the given inequality
sign is < or >, and a solid line if the
inequality symbol is ≤ or ≥.
Slide 8.5- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES
Step 3. The graph in Step 2 will divide the
plane into two regions. Select a test
point in the plane. Be sure that the test
point does not lie on the graph of the
equation in Step 1.
Step 4. (i) If the coordinates of the test point
satisfy the given inequality, then so do
all the points of the region that contains
Slide 8.5- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES
the test point. Shade the region that contains
the test point.
(ii) If the coordinates of the test point do not
satisfy the given inequality, shade the region
that does not contain the test point.
The shaded region (including the boundary if it
is a solid curve) is the graph of the inequality.
Slide 8.5- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1Using Substitution to Solve a Nonlinear System
Solve the system of equations by the substitution method.
Solution
Step 1 Solve for one variable. Express y in terms of x in equation (2).
4x y 3 (1)
x2 y 1 (2)
y x2 1
Step 2 Substitute. Substitute x2 +1 for y in equation (1).
Slide 8.5- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1Using Substitution to Solve a Nonlinear System
Solution continued
Step 3 Solve the equation resulting from step (2).
4x y 3
4x x2 1 3
4x x2 1 3
x2 4x 4 0
x 2 x 2 0
x 2 0
x 2
Slide 8.5- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1Using Substitution to Solve a Nonlinear System
Solution continued
Step 4 Back substitute. Substitute x = –2 in Equation (3) to obtain the corresponding y-value.
y x2 1
y 2 2 1
y 5
Since x = –2 and y = 5, the apparent solution set of the system is {(–2, 5)}.
Slide 8.5- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1Using Substitution to Solve a Nonlinear System
Solution continued
Step 5 Check. Replace x by –2 and y by 5 in both equations (1) and (2).
Confirm the solution with a graph.
4x y 3
4 2 5 3
8 5 3
3 3
?
?
x2 y 1
2 2 5 1
4 5 1
1 1
?
?
Slide 8.5- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1Using Substitution to Solve a Nonlinear System
Solution continued
The graphs of the line 4x + y = –3 and the parabola y = x
2 + 1 confirm that the solution set is {(–2, 5)}.
Slide 8.5- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2Using Elimination to Solve a Nonlinear System
Solve the system of equations by the elimination method.
Solution
Step 1 Adjust the coefficients. Multiply Equation (2) by –1 to eliminate x.
x2 y2 25 (1)
x2 y 5 (2)
y2 y 20Step 2
x2 y2 25 (1)
–x2 y –5 (3)
Slide 8.5- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2Using Elimination to Solve a Nonlinear System
Solution continued
Step 3 Solve the equation obtained in Step 2.y2 y 20
y2 y 20 0
y 5 y 4 0
y 5 0 or y 4 0
y 5 or y 4
Step 4 Back substitute the values in one of the original equations to solve for the other variable.
Slide 8.5- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2Using Elimination to Solve a Nonlinear System
Solution continued
(i) Substitute y = –5 in Equation (2) & solve for x.
Thus (0, –5) is a solution of the system.
x2 y 5
x2 5 5
x2 5 5
x2 0
x 0
Slide 8.5- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2Using Elimination to Solve a Nonlinear System
Solution continued
(ii) Substitute y = 4 in Equation (2) & solve for x. x2 y 5
x2 4 5
x2 4 5
x2 9
x 3
Thus (3, 4) and (–3, 4) are the solutions of the system.
Slide 8.5- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2Using Elimination to Solve a Nonlinear System
Solution continued
Step 5 Check (0, –5), (3, 4), and (–3, 4) in the equations x2 + y2 = 25 and x2 – y = 5.
02 5 2 25
25 25? 32 42 25
9 16 25
25 25
?
? 3 2 42 25
9 16 25
25 25
?
?
0 2 5 5
5 5? 32 4 5
9 4 5
5 5?
?
3 2 4 5
9 4 5
5 5
?
?
Slide 8.5- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2Using Elimination to Solve a Nonlinear System
Solution continued
The graphs of the circle x2 + y2 = 25 and the parabolay = x
2 – 5 confirm that the solution set is{(0, –5), (3, 4), (–3, 4)}.
Slide 8.5- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING A NONLINEAR INEQUALITY IN TWO VARIABLES
Step 1. Replace the inequality symbol by an
equals (=) sign.
Step 2. Sketch the graph of the corresponding
equation in Step 1. Use a dashed curve
if the given inequality sign is < or >,
and a solid line if the inequality
symbol is ≤ or ≥.
Slide 8.5- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING A NONLINEAR INEQUALITY IN TWO VARIABLES
Step 3. The graph in Step 2 will divide the
plane into two regions. Select a test
point in the plane. Be sure that the test
point does not lie on the graph of the
equation in Step 1.
Step 4. (i) If the coordinates of the test point
satisfy the given inequality, then so do
all the points of the region that contains
Slide 8.5- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING A NONLINEAR INEQUALITY IN TWO VARIABLES
the test point. Shade the region that contains
the test point.
(ii) If the coordinates of the test point do not
satisfy the given inequality, shade the region
that does not contain the test point.
The shaded region (including the boundary if it
is solid) is the graph of the given inequality.
Slide 8.5- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Solving a Nonlinear System of Inequalities
Graph the solution set of the following system of
Solution
Graph each inequality separately in the same coordinate plane. Since (0, 0) is not a solution of any the corresponding equations, use (0, 0) as a test point for each inequality.
y 4 x2 (1)
y 3
2x 3 (2)
y 6x 3 (3)
inequalities:
Slide 8.5- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Solving a Nonlinear System of Inequalities
Solution continued
Step 2 Sketch as a solid curve with vertex (0, 4).
Step 3 Test (0, 0). 0 ≤ 4 – 0 is a false statement.
Step 4 Shade the region.
Step 1 y = 4 – x2
Slide 8.5- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Solving a Nonlinear System of Inequalities
Solution continued
Step 2 Sketch as a solid line through (0, –3) & (2, 0).
Step 3 Test (0, 0). 0 ≥ 0 – 3 is a true statement.
Step 4 Shade the region.
Step 1 y 3
2x 3
Slide 8.5- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Solving a Nonlinear System of Inequalities
Solution continued
Step 3 Test (0, 0). 0 ≥ 0 – 3 is a true statement.
Step 4 Shade the region.
Step 1 y = –6x – 3
Step 2 Sketch as a solid line through (0, –3)
& 1
2,0
.
Slide 8.5- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Solving a Nonlinear System of Inequalities
Solution continued
The region common to all three graphs is the graph of the solution set of the given system of inequalities.