13-1 and 13-2: Linear and Nonlinear Systems
Objectives:
1. To solve a system of
equations consisting
of a linear and
nonlinear equation
Assignment:
• P. 221: 1-30
• Challenge Problems
Objective 1
You will be able to solve a
system of equations consisting of
a linear and nonlinear equation
System of Equations
A system of equations is a collection of 2
or more equations, linear or not, with the
same variables.
The solution to a
system of equations is
the set of all points
(𝒙, 𝒚) that satisfy all the
equations in the
system.
𝑦 =− 𝑥−3
𝑦 = −𝑥 + 3
2
+6
Geometric Interpretation
Geometrically,
the solution to a
system of
equations
occurs at the
intersection of
the graphs of
the equations.
Geometric Interpretation
So a simple way to
solve a system of
equations is to:
Graph
the 1
Step
equations
Find the
points of
Step
2
intersection
Devils Haircut
Demand
Customers will demand
250 haircuts per week at
$20 each. For each $5
increase in price,
demand will decrease by
25 haircuts.
𝑓 𝑥 = −5𝑥 + 350
Supply
Price
# of Haircuts
20
15
30
55
40
115
50
195
1 2
𝑔 𝑥 =
𝑥 −𝑥−5
10
# of Haircuts
Devils Haircut
Price of Haircut
Even
though this
system has
2 solutions,
only one is
reasonable
Number of Solutions
𝑦=𝑥
𝑦 = 𝑥2 − 2
𝑦 = 2𝑥 − 3
𝑦 = 𝑥2 − 2
𝑦 = 3𝑥 − 9
𝑦 = 𝑥2 − 2
2 Solutions
1 Solution
0 Solutions
Substitution Property
In the previous activity, you solved a system of
equations by graphing, which was debatably
accurate. We can solve the same system
algebraically by using substitution.
Substitution Property of Equality
If 𝑎 = 𝑏, then 𝑎 can be substituted for 𝑏 in
any equation or expression.
Example 1
Use substitution to solve the system of
equations.
𝑦 =− 𝑥−3 2+6
𝑦 = −𝑥 + 3
Example 2
Use substitution to solve the system of
equations.
1 2
𝑥 −𝑥−5
10
𝑦 = −5𝑥 + 350
𝑦=
Example 3
Use substitution to solve the system of
equations.
𝑦 = −𝑥 2 + 18𝑥 − 29
𝑦 = 4𝑥 + 24
13-1 and 13-2: Linear and Nonlinear Systems
Objectives:
1. To solve a system of
equations consisting
of a linear and
nonlinear equation
Assignment:
• P. 221: 1-30
• Challenge Problems