The equation, **y = mx + b,** is the slope-intercept form of a straight line. Here, x and y are the coordinates of the points, m is the gradient, and b is the intercept of the y-axis. The equations of lines can be of different forms based on the information we have. Suppose the coordinates of two points are given, which forms a straight line, then the line will form a linear equation (e.g. y = x + 3, where x and y are the coordinates of the point). The general form of the equation of the straight line is given by Ax + By + C = 0, for a line.

## What is y = mx + b?

y = mx + b is the slope-intercept form of the equation of a straight line. In the equation y = mx + b, m is the slope of the line and b is the intercept. x and y represent the distance of the line from the x-axis and y-axis, respectively. The value of b is equal to y when x = 0, and m shows how steep the line is. The slope of the line is also called the gradient.

The formula to find the slope, m, of the line is given by:

m = (difference in y coordinates)/(difference in x coordinates)

\(\begin{array}{l}m = \frac{y_2-y_1}{x_2-x_1}\end{array} \)

The equation of a line passing through a point (x_{1}, y_{1}) is given by:

y – y_{1} = m(x – x_{1})

The equation of a line passing through two points (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by:

\(\begin{array}{l}\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\end{array} \)

## How To Find y = mx + b?

To find the equation of the straight line, we use the slope-intercept form, y = mx + b, where m is the slope of the line, b is the y-intercept of the line.

We can find the equation of a line in the form of y = mx + b, if the coordinates of points forming the line are known to us.

The slope of the line, m can also be written as:

m = (y-b)/x

So, the formula to find the slope of the straight line is:

m = change in y/change in x

Now, suppose we have two points on a straight line whose coordinates are (x_{1}, y_{1}) and (x_{2}, y_{2}). Thus, we can write:

y_{1} = mx_{1} + b and y_{2} = mx_{2} + b

Since, m is the ratio of change in y to change in x, thus;

\(\begin{array}{l}\frac{y_{2}-y_{1}}{x_{2}-x_{1}} =\frac{(m x_{2}+b)-(m x_{1}+b)}{x_{2}-x_{1}}\end{array} \)

\(\begin{array}{l}=\frac{m x_{2}-m x_{1}}{x_{2}-x_{1}}\end{array} \)

Taking m common, we get,

\(\begin{array}{l}=\frac{m( x_{2}- x_{1})}{x_{2}-x_{1}}\end{array} \)

= m

Hence,

\(\begin{array}{l}m = \frac{y_2-y_1}{x_2-x_1}\end{array} \)

I.e., m= Difference in y coordinates / Difference in x coordinates.

## Y = mx + b at Origin

The equation of a straight line with slope m passing through the origin (0,0) is given by:

y = mx

Hence, the y-intercept at the origin is zero.

## Solved Examples

**Example 1:**

Find the slope and y-intercept of the equation, y = 3x – 2.

**Solution: **

If we compare the given equation with y = mx + b, where m is the slope and b is the y-intercept, then we get,

Slope, m = 3

y-intercept, b = -2

**Example 2:**

What is the slope and y-intercept of the equation, y= 5x?

**Solution: **

If we compare the given equation with y = mx + b, where m is the slope and b is the y-intercept, then we get,

Slope, m = 5

y-intercept, b = 0

Here, the y-intercept is zero, which proves that the slope of the line passes through the origin.

**Example 3:**

If the slope of a straight line is 5 and the y-intercept is 3, then find the equation of the line.

**Solution: **

We know the equation of the line in slope-intercept form is given by:

y = mx + b

Given, m = 5 and b = 3.

Thus, the required equation is:

y = 5x + 3