1. Copyright © Cengage Learning. All rights reserved.9
Systems of Equations
and Inequalities
Copyright © Cengage Learning. All rights reserved.

2. Copyright © Cengage Learning. All rights reserved.
9.1
LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS
Copyright © Cengage Learning. All rights reserved.

3. What You Should Learn
Use the method of substitution to solve systems of linear equations in two and three variables.
Use the method of substitution to solve systems of nonlinear equations in two and three variables.

4. The Method of Substitution

5. The Method of Substitution

6. Solving a System of Linear Equations in Two Variables
Solve the system of equations.
x + y = 4
x – y = 2
Solution:
Solving for y in Equation 1.
y = 4 – x
Substitute y = 4 – x in Equation 2 and solve the resulting single-variable equation for x.
Equation 1
Equation 2
Solve for y in Equation 1.
Write Equation 2.

7. Example 1 – Solution x – (4 – x) = 2 x – 4 + x = 2 2x = 6 x = 3
cont’d
x – (4 – x) = 2
x – 4 + x = 2
2x = 6
x = 3
Finally, you can solve for y by back-substituting x = 3 into the equation y = 4 – x, to obtain
y = 4 – x
y = 4 – 3
Substitute 4 – x for y.
Distributive Property
Combine like terms.
Divide each side by 2.
Write revised Equation 1.
Substitute 3 for x.

8. Example 1 – Solution
cont’d
y = 1.
The solution is the ordered pair (3, 1). You can check this solution as follows.
Solve for y.

9. Solving a System of Linear Equations in Three Variables

10. Nonlinear Systems of Equations

11. Example 3 – Substitution: Two-Solution Case
Solve the system of equations.
3x2 + 4x – y = 7
2x – y = –1
Solution:
Solving for y in Equation 2 to obtain y = 2x + 1.
Substitute y = 2x + 1 into Equation 1 and solve for x.
3x2 + 4x – (2x + 1) = 7
3x2 + 2x – 1 = 7
Equation 1
Equation 2
Substitute 2x + 1 for y in Equation 1.
Simplify.

12. Example 3 – Solution 3x2 + 2x – 8 = 0 (3x – 4)(x + 2) = 0
cont’d
3x2 + 2x – 8 = 0
(3x – 4)(x + 2) = 0
Back-substituting these values of x to solve for the
corresponding values of y produces the solutions
and
Write in general form.
Factor.
Solve for x.

13. Solve systems of nonlinear equations

14. What You Should Learn
Use the method of elimination to solve systems of linear equations in two and three variables.

15. The Method of Elimination

16. The Method of Elimination

17. Solving a System of Linear Equations in Two Variables
Solve the system of linear equations.
Solution:
For this system, you can obtain coefficients that differ only in sign by multiplying Equation 2 by 4.
Equation 1
Equation 2
Write Equation 1.
Multiply Equation 2 by 4.
Add equations.
Solve for x.

18. Example 2 – Solution
cont’d
By back-substituting into Equation 1, you can solve for y.
2x – 4y = –7
–4y = –7
The solution is
Write Equation 1.
Substitute for x.
Combine like terms.
Solve for y.

19. Solving a System of Linear Equations in Three Variables

20. What You Should Learn
Sketch the graphs of inequalities in two variables.
Solve systems of inequalities.

21. The Graph of an Inequality

22. The Graph of an Inequality

23. Example 1 – Sketching the Graph of an Inequality
Sketch the graph of y  x2 – 1.
Solution:
Step 1: Change the inequality sign to an equal sign.
y = x2 – 1
Step 2: Sketch the graph.
Step 3: Test a point.
Substitute (0,0) into y  x2 – 1
Substitute (0,-2) into y  x2 – 1
We can see that the points above (or on) the graph satisfy the inequality.
Step 4: Shade the region containing the point (0,0)

24. Systems of Inequalities

25. Example 4 – Solving a System of Inequalities
Sketch the graph (and label the vertices) of the solution set of the system.
x – y < 2
x > –2
y  3
Inequality 1
Inequality 2
Inequality 3

26. Example 1 – Sketching the Graph of an Inequality
Solution:
Step 1: Change the inequality sign to an equal sign.
y = x-2
x = –2
y = 3
Step 2: Sketch the graph.
Step 3: Test a point for each inequality.
Step 4: Shade the region.
Step 5: The overlapping region is the solution.

27. Example 4 – Solution To find the vertices of the region, solve:
cont’d
To find the vertices of the region, solve:
Vertex A: x – y = 2
x = –2
A(–2, –4)
Vertex B: x – y = 2
y = 3
B(5, 3)
Vertex C: x = –2
y = 3
C(–2, 3)