1. Nonlinear Systems of Equations
Objective: To solve a nonlinear system of equations.

2. Review – What? System – 2 or more equations together
Solution of system – any ordered pair that makes all equations true
Possible solutions:
One point
More than one point
No solution
Infinite solutions

3. Review - How?
What methods have we used to solve linear systems of equations?
Graphing
Substitution
Elimination

4. Review Steps for using SUBSTITUTION
Solve one equation for one variable. (Hint: Look for an equation already solved for a variable or for a variable with a coefficient of 1 or -1.)
Substitute into the other equation.
Solve this equation to find a value for the variable.
Substitute again to find the value of the other variable.
Check.

5. Review Solve using Substitution.
2x – y = 6
y = 5x

6. Review STEPS for ELIMINATION

7. Review Solve using elimination.
3x + 5y = 11
2x + 3y = 7

8. What’s New?
A non-linear system is one in which one or more of the equations has a graph that is not a line.
With non-linear systems, the solution could be one or more points of intersection or no point of intersection.
We’ll solve non-linear systems using substitution or elimination.
A graph of the system will show the points of intersection.

9. Solve the following system of equations:
An Example…
Solve the following system of equations:

10. An Example… We use the substitution method.
First, we solve equation (2) for y.

11. An Example…
Next, we substitute y = 2x  3 in equation (1) and solve for x:

12. An Example… … x = 0 x = 12 / 5 y = 2x  3 y = 2(0)  3 y = 3
Now, we substitute these numbers for x in equation (2) and solve for y.
x = x = 12 / 5
y = 2x  3
y = 2(0)  3
y = 3
SOLUTIONS
(0, 3) and

13. An Example…
Visualizing the Solution
Check: (0, 3)
Check:

14. See Example 1, page 747 Check Point 1: Solve by substitution.
x2 = y – 1
4x – y = -1

15. Example 2 Check Point 2: Solve by substitution. x + 2y = 0

16. Another example to watch…
Solve the following system of equations:
xy = 4
3x + 2y = 10

17. Solve xy = 4 for y.
Substitute into 3x + 2y = 10.

18. Use the quadratic formula (or factor)
to solve:

19. Substitute values of x to find y.
3x + 2y = 10
x = 4/ x = 2
The solutions are
(4/3, 3) and (2, 2).
Visualizing the Solution

20. Need to watch another one?
Solve the system of equations:

21. Solve by elimination.
Multiply equation (1) by 2 and add to eliminate the y2 term.

22. Substituting x = 1 in equation (2) gives us:
x = x = -1
The possible solutions are
(1, 3), (1, 3), (1, 3) and (1, 3).

23. Visualizing the Solution
All four pairs check, so they are the solutions.

24. Example 3 Check Point 3: Solve by elimination. 3x2 + 2y2 = 35

25. Example 4
Check Point 4: Solve by elimination.
y = x2 + 5
x2 + y2 = 25