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Wize High School Algebra I Textbook (Common Core) > Relations & Functions

Linear Relations

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Linear Relations

A linear relation is where the dependent variable changes by an amount that is proportional to the change in the independent variable.

*This means that the rate of change (how fast the value of the variables are changing) is constant.

Example
A certain YouTuber gets paid $0.01 per view for her videos. Is the relation between the number of views and the total amount this YouTuber gets paid a linear relation?

Independent variable:
the number of views (v)
Dependent variable:
the total amount of pay in $ (p)

How do we tell if a relation is linear?

Table of Values The first differences in the table of values of a linear relation is constant


Graph The graph of a linear relation is a straight line


Equation The equation of a linear relation has a degree of 1
p=0.01 vp=0.01\ v
The independent variable vv is raised to an exponent of 1, so the equation has a degree of 1.


Answer
Therefore, this is a linear relation.
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Example: Linear Relations

A company pays a base rate of $20 per day plus $10 per hour worked.
a) Identify the independent and dependent variables in this situation.

Independent variable: the number of hours worked.
Dependent variable: the amount of total pay.

b) Create a table of values for this situation.


c) Create the graph based on the table of values you create in part b).


d) Write the equation that represents this situation.

Let pp represent the total amount of pay in $.
Let tt represent the number of hours worked.

The equation is p=20+10tp=20+10t or p=10t+20p=10t+20.

e) Determine whether the relation between the number of hours worked and the amount of daily pay is linear or not.

This is a linear relation:
  • Table of value: the first differences are constant
  • The graph: the graph is a straight line
  • The equation: the equation has a degree of 1

Practice: Linear Relations

Select all of the following that represent a linear relation.


Practice: Linear Relations

Create an equation for each of the following scenarios.

a) Linda's monthly salary is calculated based on a base pay of $2,000 plus a 10% commision on the total amount of sales she does for the month.

b) The weight of my German Shepard dog when he was born was about 2 pounds. Then for the first 18 months of his life, he gained approximately 4.5 pounds per month.

c) It takes approximately 400 hours for me to paint Billy Ronka's factory by myself, but for each additional helper I get, it reduces the total paint time by 6 hours.