**Determining rate of change and initial value :**

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.

Hence, 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.

After having gone through the stuff given above, we hope that the students would have understood, how to determine rate of change and initial value

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