WRITING LINEAR EQUATIONS FROM A TABLE

Example 1 :

The table shows the temperature of a fish tank during an experiment. Write the appropriate linear equation to find the temperature at any time.

Solution :

Step 1 :

Notice that the change in the temperature is the same for each increase of 1 hour in time. So, the relationship is linear.

Step 2 :

Let x stand for time and y stand for temperature. 

Choose any two points in the form (x, y), from the table to find the slope :

For example, let us choose (0, 82) and (1, 80).

Use the slope formula. 

m  =  (y₂ - y₁) / (x₂ - x₁)

Substitute : 

(x₁, y₁)  =  (0, 82)

(x₂, y₂)  =  (1, 80)

Then, 

m  =  (80 - 82) / (1 - 0)

m  =  -2 / 1

m  =  -2

Step 3 : 

Find the y-intercept using the slope and any point from the table.

Slope-intercept form equation of a line :

y  =  mx + b

Substitute  m = -2, and (x, y) = (0, 82).

82  =  -2(0) + b

82  =  0 + b

82  =  b

Step 4 : 

Now, substitute m = -2 and b = 82 in slope-intercept form equation of a line.

y  =  mx + b

y  =  -2x + 82

Example 2 :

Elizabeth’s cell phone plan lets her choose how many minutes are included each month. The table shows the plan’s monthly cost y for a given number of included minutes x. Write an equation in slope-intercept form to represent the situation.

Solution :

Step 1 :

Notice that the change in cost is the same for each increase of 100 minutes. So, the relationship is linear.

Step 2 : 

Choose any two points in the form (x, y), from the table to find the slope :

For example, let us choose (100, 14) and (200, 20).

Use the slope formula. 

m  =  (y₂ - y₁) / (x₂ - x₁)

Substitute : 

(x₁, y₁)  =  (100, 14)

(x₂, y₂)  =  (200, 20)

Then,

m  =  (20 - 14) / (200 - 100)

m  =  6 / 100

m  =  0.06

Step 3 : 

Find the y-intercept using the slope and any point from the table.

Slope-intercept form equation of a line :

y  =  mx + b

Substitute  m  =  0.06, and (x, y)  =  (100, 14).

14  =  0.06(100) + b

14  =  6 + b

8  =  b

Step 4 : 

Now, substitute m = 0.06 and b = 8 in slope-intercept form equation of a line.

y  =  mx + b

y  =  0.06x + 8

Example 3 : 

A salesperson receives a weekly salary plus a commission for each computer sold. The table shows the total pay, y, and the number of computers sold, x. Write an equation in slope-intercept form to represent this situation.

Solution :

Step 1 :

Notice that the change in total pay is the same for increase in sales of every 2 computers. So, the relationship is linear.

Step 2 :

Choose any two points in the form (x, y), from the table to find the slope :

For example, let us choose (4, 550) and (6, 700).

Use the slope formula. 

m  =  (y₂ - y₁) / (x₂ - x₁)

Substitute : 

(x₁, y₁)  =  (4, 550)

(x₂, y₂)  =  (6, 700)

Then, 

m  =  (700 - 550) / (6 - 4)

m  =  150 / 2

m  =  75

Step 3 : 

Find the y-intercept using the slope and any point from the table.

Slope-intercept form equation of a line :

y  =  mx + b

Substitute m  =  75, and (x, y)  =  (4, 550).

550  =  75(4) + b

550  =  300 + b

250  =  b

Step 4 : 

Now, substitute m = 75 and b = 250 in slope-intercept form equation of a line.

y  =  mx + b

y  =  75x + 250

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