**Writing linear equations from situations and graphs :**

In this section, we are going to see, how to write linear equations from the given situations and graphs in slope-intercept form.

The slope-intercept form of the equation is y = mx + b, where "m" is slope and "b" is y-intercept. And y-intercept is nothing but the point at which the line cuts y-axis.

For example, if the line cuts y-axis at (0, -4), then the y-intercept is -4.

**Example 1 : **

The rent charged for space in an office building is a linear relationship related to the size of the space rented.At west main street office rentals, $750 rent charged for 600 square feet of space and $1150 rent charged for 900 square feet of space. Write an equation in slope-intercept form for the rent at West Main Street Office Rentals.

**Solution : **

**Step 1 :**

Identify the independent and dependent variables.

The independent variable (x) is the square footage of floor space.

The dependent variable (y) is the monthly rent.

**Step 2 :**

Write the information given in the problem as ordered pairs.

The rent for 600 square feet of floor space is $750 :

(600, 750)

The rent for 900 square feet of floor space is $1150 :

(900, 1150)

**Step 3 : **

Find the slope.

m = (y₂ - y₁) / (x₂ - x₁)

Substitute (600, 750) for (x₁, y₁) and (900, 1150) for (x₂, y₂).

m = (1150 - 750) / (900 - 600)

m = 400 / 300

m = 4/3

**Step 4 : **

Find the y-intercept.

Use the slope 4/3 and one of the ordered pairs (600, 750).

Slope-intercept form :

y = mx + b

Plug m = 4/3, x = 600 and y = 750.

750 = (4/3)(600) + b

750 = (4)(200) + b

750 = 800 + b

-50 = b

**Step 5 : **

Substitute the slope and y-intercept.

Slope-intercept form

y = mx + b

Plug m = 4/3 and b = -50

y = (4/3)x + (-50)

y = (4/3)x - 50

Let us look at the next example on "Writing linear equations from situations and graphs".

**Example 2 : **

Hari’s weekly allowance varies depending on the number of chores he does. He received $16 in allowance the week he did 12 chores, and $14 in allowance the week he did 8 chores. Write an equation for his allowance in slope-intercept form.

**Solution : **

**Step 1 :**

Identify the independent and dependent variables.

The independent variable (x) is number of chores Hari does per week

The dependent variable (y) is the allowance he receives per week.

**Step 2 :**

Write the information given in the problem as ordered pairs.

For 12 chores, he receives $16 allowance :

(12, 16)

For 8 chores, he receives $14 allowance :

(8, 14)

**Step 3 : **

Find the slope.

m = (y₂ - y₁) / (x₂ - x₁)

Substitute (12, 16) for (x₁, y₁) and (8, 14) for (x₂, y₂).

m = (14 - 16) / (8 - 12)

m = (-2) / (-4)

m = 1/2

m = 0.5

**Step 4 : **

Find the y-intercept.

Use the slope 0.5 and one of the ordered pairs (8, 14).

Slope-intercept form :

y = mx + b

Plug m = 0.5, x = 8 and y = 14.

14 = (0.5)(8) + b

14 = 4 + b

10 = b

**Step 5 : **

Substitute the slope and y-intercept.

Slope-intercept form

y = mx + b

Plug m = 0.5 and b = 10

y = 0.5x + 10

Let us look at the next example on "Writing linear equations from situations and graphs".

**Example 3 : **

A video club charges a one-time membership fee plus a rental fee for each DVD borrowed. Use the graph to write an equation in slope-intercept form to represent the amount spent, y, on x DVD rentals

**Solution : **

**Step 1 :**

Choose two points on the graph, (x₁, y₁) and (x₂, y₂), to find the slope.

Find the ratio between change in y-values and change in x-

m = (y₂ - y₁) / (x₂ - x₁)

Substitute (0, 8) for (x₁, y₁) and (8, 18) for (x₂, y₂).

m = (18 - 8) / (8 - 0)

m = 10 / 8

m = 1.25

**Step 2 : **

Read the y-intercept from the graph. That is, the point at which the line cuts y-axis.

The y-intercept is 8.

b = 8

**Step 3 : **

Let us use slope (m) and y-intercept (b) values to write an equation in slope-intercept form.

y = mx + b (Slope-intercept form)

Plug m = 1.25 and b = 8.

y = 1.25x + 8

Let us look at the next example on "Writing linear equations from situations and graphs".

**Example 4 : **

The cash register subtracts $2.50 from a $25 Coffee Café gift card for every medium coffee the customer buys. Use the graph to write an equation in slope-intercept form to represent this situation.

**Solution : **

**Step 1 :**

Choose two points on the graph, (x₁, y₁) and (x₂, y₂), to find the slope.

Find the ratio between change in y-values and change in x-

m = (y₂ - y₁) / (x₂ - x₁)

Substitute (0, 25) for (x₁, y₁) and (10, 0) for (x₂, y₂).

m = (0 - 25) / (10 - 0)

m = -25 / 10

m = -2.5

**Step 2 : **

Read the y-intercept from the graph. That is, the point at which the line cuts y-axis.

The y-intercept is 25.

**Step 3 : **

Let us use slope (m) and y-intercept (b) values to write an equation in slope-intercept form.

y = -2.5x + 25 (Slope-intercept form)

Plug m = -2.5 and b = 25.

y = -2.5x + 25

After having gone through the stuff given above, we hope that the students would have understood "Writing linear equations from situations and graphs".

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