# DERIVING THE SLOPE INTERCEPT FORM OF AN EQUATION

## About "Deriving the slope intercept form of an equation"

Deriving the slope intercept form of an equation :

In this section, we are going to see, how to derive the slope intercept form of an equation.

## Deriving the slope intercept form of an equation

Step 1 :

Let L be a line with slope m and y-intercept b. Circle the point that must be on the line. Justify your choice.

(b, 0)   (0, b)   (0, m)   (m, 0)

The coordinate of x is 0 in the point that includes the y-intercept.

Step 2 :

Recall that slope is the ratio of change in y to change in x. Complete the equation for the slope m of the line using the y-intercept (0, b), and another point (x, y) on the line.

Slope m  =  change in y-values / change in x-values

Slope m  =  (y - b) / (x - 0)

Slope m  =  (y - b) / x

Step 3 :

In an equation of a line, we often want y by itself on one side of the equation. Solve the equation from Step 2 for y.

m  =  (y - b) / x

Multiply both sides by x

m.x  =  [(y - b) / x].x

mx  =  y - b

Add b to both sides of the equation.

mx + b  =  (y - b) + b

mx + b  =  y

Write the equation with y on the left side.

y  =  mx + b

## Reflect

Critical thinking : Write the equation of a line with slope m that passes  through the origin. Explain your reasoning.

y = mx

Because the origin is on the y-axis, the graph crosses the y-axis at (0, 0). So, the y-intercept b is 0, and y  =  mx + b becomes y  =  mx.

## Slope-intercept form equation of a line - Examples

Example 1 :

A line is passing through the points (2, 3) and (0, 4). Find the equation of the line in slope intercept form.

Solution :

Step 1 :

Fine the slope of the line using the points (2, 3) and (0, 4).

Slope  m  =  change in y-values / change in x-values

Slope  m  =  (4-3) / (0-3)

Slope  m  =  1 / (-3)

Slope  m  =  -1/3

Step 2 :

In the point (0, 4), x-coordinate is zero. So, the line intersects y-axis at this point.

Since the y-coordinate at this point is 4, y-intercept is 4.

Hence, the equation of the line is y  =  (-1/3)x + 4.

Example 2 :

A line is passing through the points (0, 0) and (-1, -8). Find the equation of the line in slope intercept form.

Solution :

Step 1 :

Fine the slope of the line using the points (0, 0) and (-1, -8).

Slope  m  =  change in y-values / change in x-values

Slope  m  =  (-8-0) / (-1-0)

Slope  m  =  -8 / (-1)

Slope  m  =  8

Step 2 :

Since the line is passing through the origin (0,0), there is no y-intercept or y-intercept = 0.

Hence, the equation of the line is y  =  8x.

After having gone through the stuff given above, we hope that the students would have understood, how to derive the slope intercept form of an equation.

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