Chapter 7.3 covers methods for solving systems of nonlinear equations using substitution and elimination, as well as graphing techniques. It emphasizes the importance of mastering these methods for finding solutions, including scenarios where answers may be irrational. Assignments and quizzes will test knowledge of these various techniques.

1. 7.3 Systems of Nonlinear
Equations
Chapter 7 Systems of Equations and Inequalities

2. Concepts and Objectives
⚫ The objectives for this section are
⚫ Solve a system of nonlinear equations using
substitution.
⚫ Solve a system of nonlinear equations using
elimination.
⚫ Graph a nonlinear inequality.
⚫ Graph a system of nonlinear inequalities.

3. Nonlinear Systems
⚫ A nonlinear system is one in which at least one equation
is not linear.
⚫ You can solve a nonlinear system by
⚫ Elimination – transform one or both equations so that
you can eliminate one of the variables by combining
the equations together
⚫ This can be tricky if you can’t match up the exponents
on the variables as well.
⚫ Substitution – solve one equation for one variable
and substitute it into the other equation(s)
⚫ Graphing

4. Solving by Elimination
⚫ Example: Solve the system and write your solution as a
set of ordered pairs.
 + =

− =

2 2
2 2
2 5 98
2 2
x y
x y

5. Solving by Elimination
⚫ Example: Solve the system and write your solution as a
set of ordered pairs.
Because the coefficients and exponents of x are the
same, this is a good candidate for elimination. To solve,
multiply the second equation by ‒1 and combine the
two equations.
 + =

− =

2 2
2 2
2 5 98
2 2
x y
x y

6. Solving by Elimination
⚫ Example: Solve the system and write your solution as a
set of ordered pairs.
Now plug in y in one of the equations and solve for x.
 + =

− =

2 2
2 2
2 5 98
2 2
x y
x y
+ =
− + = −
2 2
2 2
2 5 98
2 2
x y
x y
 ‒1
=
2
6 96
y
=
2
16
y
= 4
y

7. Solving by Elimination
⚫ Example: Solve the system and write your solution as a
set of ordered pairs.
− =
2
2 16 2
x (we already know y2 = 16)
=
2
2 18
x
=
2
9
x
= 3
x
( ) ( ) ( ) ( )
 
− − − −
3,4 , 3, 4 , 3,4 , 3, 4

8. Solving by Elimination
⚫ The graph of the system shows why there are four
solutions:
+ =
2 2
2 5 98
x y
− =
2 2
2 2
x y

9. Solving by Substitution
⚫ Example: Solve by substitution and write the solution as
a set of ordered pairs.
 + =

− = −

2 2
25
5
x y
y x

10. Solving by Substitution
⚫ Example: Solve by substitution and write the solution as
a set of ordered pairs.
Solve the second equation for y and substitute into the
first equation:
 + =

− = −

2 2
25
5
x y
y x
−
= 5
y x ( ) =
−
+
2
2
5
5 2
x
x

11. Solving by Substitution
⚫ Example: Solve by substitution and write the solution as
a set of ordered pairs.
Plug each value of x back in and solve for y.
( )
+ − =
2
2
5 25
x x
+ − + =
2 2
10 25 25
x x x
− =
2
2 10 0
x x
( )
− =
2 5 0
x x
= 0,5
x

12. Solving by Substitution
⚫ Example: Solve by substitution and write the solution as
a set of ordered pairs.
= −
= −
0 5
5
y = −
=
5 5
0
y
( ) ( )
 
−
0, 5 , 5,0
+ =
2 2
25
x y
− = −5
y x

13. Solving by Graphing
⚫ Example: Solve the system by graphing and write your
solution as a set of ordered pairs.
Open desmos.com/calculator (or use the app). Type the
two equations in. On the graph, click on the intersection
point(s). The program should display the coordinates.
Write that as your solution.
− =
+ = −
2
4
2
x y
x y

14. Solving by Graphing
⚫ Example: Solve the system by graphing and write your
solution as a set of ordered pairs.
− =
+ = −
2
4
2
x y
x y
The solution is
( ) ( )
 
2,0 , 1, 3
− −

15. Final Note
⚫ Although I have shown you how to find solutions by
graphing, I expect you to be able to solve by any method.
Some of the questions I ask on the classwork checks,
quizzes, and tests will be structured so that you will
need to know the other methods as well. This is
especially important if your answer is an irrational
number (such as , for example) – Desmos will only
give you a decimal approximation.
5 2

16. Classwork
⚫ College Algebra 2e
⚫ 7.3: 6-20 (even); 7.2: 38-44 (even); 7.1: 62-76 (even)
⚫ 7.3 Classwork Check
⚫ Quiz 7.2