Activity
Linear models can also be found by hand in some situations when the data follows a linear pattern since the slope remains the same between data points.
Linear Models from Tables
For 1 - 4, use the scenario and table below:
The table below shows the number of songs Marcus will have in his collection as he adds new songs each month.
Number of months, t 0 1 2 3 Number of songs, N 200 215 230 245
What is the initial amount of songs, or the -intercept of this situation?
200
What is the slope, or rate of change?
15
Write the equation of the linear model that represents the situation in slope-intercept form.
Compare your answer:
How many songs will Marcus have in 8 months?
320
Compare your answer:
For 5 - 8, use the situation and table below:
A new plant food is introduced to a young tree to test its effect on the height of the tree. The table shows the height of the tree, H, in feet months since measurements began.
Number of months, x 2 4 6 8 Height in feet, H 13.5 14.5 15.5 16.5
What is the initial height of the tree, or the -intercept, when measurements began?
(Hint: Work each row backwards to where .)
12.5
What is the slope, or rate of change?
Write the equation of the linear model that represents the situation in slope-intercept form.
Compare your answer:
+12.5\)
What will the height of the tree be in 14 months?
19.5
Compare your answer:
Linear Models from Graphs
For 9 - 12, use the situation and graph below:
The graph models the cost in dollars, , of renting a tent at a campground for n nights.
![]()
What is the -intercept of the graph?
30
What is the slope, or rate of change?
10
Write the equation of the linear model that represents the situation in slope-intercept form.
Compare your answer:
How much does it cost to rent a tent for 5 nights?
Compare your answer:
Self Check
The table below shows the revenue for the number of pizzas sold at a restaurant.
Number of Pizzas | 20 | 30 | 40 | 50 |
Revenue | 110 | 160 | 210 | 260 |
Which linear model could be used to predict the revenue if 4000 pizzas were sold?
Additional Resources
Writing Linear Equations From Tables and Graphs
Linear Equations from Tables
The number of texts a teen sends, , in days, , is shown in the table below.
Days, | 1 | 2 | 3 | 4 |
Number of texts, | 65 | 130 | 195 | 260 |
Write an equation in slope-intercept form to represent the situation then predict the number of texts sent in 8 days.
Step 1 - Find the -intercept.
Since the value when is not present, work backwards to .
If was present, then since as each x-value increases by 1, the -values increase by 65.
Step 2 - Find the slope of the situation.
Find the rate of change. The change in y is 65 and the change in is 1, so the slope is 65
Step 3 -Write the equation in slope-intercept form.
Step 4 -Make a prediction.
Substitute into the equation.
After 8 days, the teen would have sent 520 texts.
Linear Equations from Graphs
The cost, C, for the number of days, , a dog spends at doggie daycare is shown in the graph below.
Write an equation in slope-intercept form to represent the situation then predict the cost of a dog staying at doggie daycare after 7 days.
Step 1 - Find the -intercept.
Since the value when is not present, work backwards to .
If was present, then since as each -value increases by 1, the -values increase by 35.
Step 2 - Find the slope of the situation.
Find the rate of change. The change in is 35 and the change in is 1, so the slope is 35
Step 3 -Write the equation in slope-intercept form.
Step 4 -Make a prediction.
Substitute into the equation.
After 7 days, the cost of doggie daycare would be $245.
Try it
Try It-Writing Linear Equations From Tables and Graphs
Naomi is a professional painter. The table below shows how many square feet, , she can paint in hours.
Time in hours, | 1 | 2 | 3 | 4 |
Number of square feet, | 120 | 240 | 360 | 480 |
How many square feet can Naomi paint in 8 hours?
Compare your answer:
960
Step 1 - Find the -intercept.
Since the value when is not present, work backwards to .
If was present, then since as each -value increases by 1, the -values increase by 120.
Step 2 - Find the slope of the situation.
Find the rate of change. The change in is 120 and the change in is 1, so the slope is 120
Step 3 -Write the equation in slope-intercept form.
Step 4 -Make a prediction.
Substitute into the equation.
Naomi can paint 960 square feet in 8 hours.