The linear equation shown is written in the slope-intercept form of an equation. Its graph is a line with slope m and y-intercept b.
A linear relationship has a constant rate of change. You can find the rate of change m and the initial value b for a linear situation from a table of values.
Example 1 :
Find the slope and y-intercept of the line represented by the table. And also find the equation of the line in slope-intercept form.
Solution :
Step 1 :
Confirm that the rate of change is constant.
Change in y-value / change in x-value :
= (32 - 22)/(4 - 2) = 10/2 = 5
= (42 - 32)/(6 - 4) = 10/2 = 5
= (52 - 42)/(8 - 6) = 10/2 = 5
The rate of change is a constant and it is 5.
So, the slope is 5.
Step 2 :
Find the initial value y. That is, the value of y when x = 0.
Work backward from x = 2 to x = 0 to find the initial value.
The initial value is 12. That is, y-intercept is 12.
So, the equation of the line is
y = 5x + 12
Example 2 :
A phone salesperson is paid a minimum weekly salary and a commission for each phone sold, as shown below. Confirm that the relationship is linear and give the constant rate of change and the initial value.
No. of phones sold
10
20
30
40
Weekly income ($)
$480
$630
$780
$930
Solution :
Step 1 :
Confirm that the rate of change is constant.
Phones Sold (10 to 20) :
Change in income/Change in phones sold is
= (630 - 480)/(20 - 10)
= 150/10
= 15
Phones Sold (20 to 30) :
Change in income/Change in phones sold is
= (780 - 630)/(30 - 20)
= 150/10
= 15
Phones Sold (30 to 40) :
Change in income/Change in phones sold is
= (930 - 780)/(40 - 30)
= 150/10
= 15
The rate of change is a constant and it is 15.
The salesperson receives a $15 commission for each phone sold.
Step 2 :
Find the initial value when the number of phones sold is 0.
Work backward from x = 10 to x = 0 to find the initial value.
The initial value is $330. The salesperson receives a salary of $330 each week before commissions.
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