1. LIAL HORNSBY SCHNEIDER
COLLEGE ALGEBRA
LIAL
HORNSBY
SCHNEIDER

2. 2.6 Nonlinear Systems of Equations
Solving Nonlinear Systems with Real Solutions
Solving Nonlinear Equations with Nonreal Complex Solutions

3. Solve the system. Solution SOLVING A NONLINEAR SYSTEMS BY SUBSTITUTION
Example 1
Solve the system.
(1)
(2)
Solution
When one of the equations in a nonlinear system is linear, it is usually best to begin by solving the linear equation for one of the variables.
Solve equation (2) for y.
Substitute this result for y in equation (1).

4. SOLVING A NONLINEAR SYSTEMS BY SUBSTITUTION Example 1
(1)
Let y = – 2 – x.
Distributive property
Standard form
Factor.
Zero factor property

5. SOLVING A NONLINEAR SYSTEMS BY SUBSTITUTION
Example 1
Substituting – 2 for x in equation (2) gives y = 0. If x = 1, then y = – 3. The solution set of the given system is {(– 2, 0),(1, – 3)}.

6. Visualizing Graphs
Visualizing the types of graphs involved in a nonlinear system helps predict the possible numbers of ordered pairs of real numbers that may be in the solution set of the system.

7. Solve the system. Solution
SOLVING A NONLINEAR SYSTEM BY ELIMINATION
Example 2
Solve the system.
(1)
(2)
Solution
The graph of equation (1) is a circle and, as we will learn in the future units, the graph of equation (2) is a hyperbola. These graphs may intersect in 0, 1, 2, 3, or 4 points. We add to eliminate y2.

8. Remember to find both square roots.
SOLVING A NONLINEAR SYSTEM BY ELIMINATION
Example 2
(1)
(2)
Add.
Divide by 3.
Remember to find both square roots.
Square root property

9. Find y by substituting back into equation (1).
SOLVING A NONLINEAR SYSTEM BY ELIMINATION
Example 2
Find y by substituting back into equation (1).
If x = 2, then
If x = − 2, then
The solutions are (2, 0) and (– 2, 0) so the solution set is {(2, 0), (– 2, 0)}.

10. Solve the system. Solution Example 3
SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS
Example 3
Solve the system.
(1)
(2)
Solution
(1)
Multiply (2) by – 1
Add (3)
Solve for y(x ≠ 0).

11. SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS
Example 3
Now substitute for y in either equation (1) or (2). We use equation (2).
Let y = in (2).
Multiply and square.
Multiply x2 to clear fractions.

12. Example 3 SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS
This equation is in quadratic form.
Subtract 6x2.
Factor.
Zero factor property.
Square root property;
For each equation, include both square roots.

13. SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS
Example 3
Substitute these x-values into equation (4) to find corresponding values of y.
If , then
If , then
If , then
If , then

14. The solution set of the system is
SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS
Example 3
The solution set of the system is
If , then
If , then
If , then
If , then

15. Solve the system. Solution Example 4
SOLVING A NONLINEAR SYSTEM WITH AN ABSOLUTE VALUE EQUATION
Example 4
Solve the system.
(1)
(2)
Solution
Use the substitution method. Solving equation (2) for x gives
(3)
Since x  0 for all x, 4 – y  0 and thus y  4. In equation (1), the first term is x2, which is the same as x2.

16. Remember the middle term.
SOLVING A NONLINEAR SYSTEM WITH AN ABSOLUTE VALUE EQUATION
Example 4
Therefore,
Substitute 4 – y in (1).
Square the binomial.
Combine terms.
Remember the middle term.
Factor.
Zero factor property.

17. SOLVING A NONLINEAR SYSTEM WITH AN ABSOLUTE VALUE EQUATION
Example 4
To solve for the corresponding values of x, use either equation (1) or (2). We use equation (1).
If y = 0, then
If y = 4, then

18. SOLVING A NONLINEAR SYSTEM WITH AN ABSOLUTE VALUE EQUATION
Example 4
The solution set, {(4,0), (– 4, 0), (0, 4)}, includes the points of intersection shown in the graph.

19. Solve the system. Solution Example 5
SOLVING A NONLINEAR SYSTEM WITH NONREAL COMPLEX NUMBERS IN ITS SOLUTIONS
Example 5
Solve the system.
(1)
(2)
Solution
Multiply (1) by – 3 .
(2)
Add.
Square root property

20. To find the corresponding values of y, substitute into equation (1).
SOLVING A NONLINEAR SYSTEM WITH NONREAL COMPLEX NUMBERS IN ITS SOLUTIONS
Example 5
To find the corresponding values of y, substitute into equation (1).
If x = 2i, then
If x = – 2i, then
i2 = – 4

21. SOLVING A NONLINEAR SYSTEM WITH NONREAL COMPLEX NUMBERS IN ITS SOLUTIONS
Example 5
Checking the solutions in the given system shows that the solution set is
Note that these solutions with nonreal complex number components do not appear as intersection points on the graph of the system.