Straight lines can be expressed in two types of equation – one is called the general form and the other form is called the slope intercept form. The general form is $ax + by + c$ = $0$ such that when $a$ = $0$, the line is horizontal and when $b$ = $0$, then the line is vertical. Slope intercept form is $y$ = $mx + c$, where ‘$m$’ denotes the slope or gradient of the line and ‘$c$’ denotes the $y$-intercept, $x$ is the distance covered left and right to the $y$ axis and ‘$y$’ denotes the distance covered up and down the $x$ axis. Slope or gradient is defined as the ratio of change in $y$ over change in $x$.

## Definition

Slope intercept form is the most common method of expressing the equation of a straight line. To form the slope intercept form of a line, we need to know two things – first is ‘$m$’ which mean slope of a line and $b$ which is called they intercept of the straight line. The formula of slope intercept form of a straight line is $y$ = $mx + b$.

## Formula

Linear equation in slope intercept form is written as $y$ = $mx + b$, where ‘$m$’ stands for the slope of the line, ‘$b$’ stands for Formula $y$ intercept. By $y$ intercept we mean the coordinate of the point or the value of $y$ at which it cuts the $y$ axis. In the slope intercept form we plugin the values of $m$ and $b$ and form the equation in $x$ and $y$. The same slope intercept form of linear equation could be checked by plugging in the values of the coordinate of any point which is lying on the line. If the coordinates of any point lying on the line satisfies the slope intercept form of the equation created, then it is said to be correctly formed.

Equation of slope intercept form for vertical line – The slope intercept form of vertical line is $x$ = $b$. The slope ‘$m$’ of vertical line is undefined and the $x$ coordinate of any point lying on that line is constant, therefore whatever is the value of $x$, it becomes the same as b. The vertical line goes straight up and down.

Equation of slope intercept form for horizontal line – The slope intercept form of horizontal line is $y$ = $b$. The slope ‘$m$’ of horizontal line is zero and the $y$ coordinate of any point lying on that line is constant, therefore whatever is the value of $y$, it becomes the same as $b$. On plugging $m$ = $0$ in the standard form $y$ = $0 \times x + b,\ y$ = $b$. The vertical line goes straight left and right.

## Slope Intercept form of a Line

The Slope – intercept form of a line is $y$ = $mx + b$, $m$ stands for the slope and ‘$b$’ the $y$ intercept. Slope is defined as the ratio of difference in $y$ to difference in $x$ values. Such that if any two points lying on a straight line has coordinates $(x_{1},\ y_{1})$ and $(x_{2},\ y_{2})$, then slope of that line would be $\frac{(y_{2} – y_{1})}{(x_{2} – x_{1})}$. The slope of a line is also known as the gradient of a straight line. It is the measure of steepness of a straight line. It is also the measure of degree by how much it deviates away from the $x$ axis or horizontal line. If the angle of deviation from the $x$ axis is theta, then the slope of the line would be $m$ = tan (theta). ‘$b$’ of slope intercept form is called the $y$ intercept meaning the value of $y$ coordinate at which it crosses the $y$ axis. Once we find out the values of slope ‘$m$’ and $y$ intercept ‘$b$’, we plugin those values in the slope intercept equation of the straight line $y$ = $mx + b$ and form the linear equation in $x$ and $y$ of the straight line in slope intercept form.

## Slope intercept form Graph

From the linear equation of straight line given in slope intercept form $y$ = $mx + b$, we get to know the values of slope ‘$m$’ and $y$ intercept ‘$b$’. Using these two values of slope and $y$ intercept we are able to plot points lying on the straight line and then join those points to draw the straight line. Let us illustrate the graphing with the help of an example
Example: $y$ = $\frac{3}{4x}$ – $5$

From the given equation we get to know the slope $(m)$ = $\frac{3}{4}$ and $y$ coordinate of the point at which it crosses the $y$ axis is $-5$. Steps to be followed while plotting the points are as follows:

Step 1: The $y$ intercept is considered to be the first point to plot, that is $(0, -5)$.

Step 2: Next we consider the slope $\frac{3}{4}$, which is move up by $3$ divisions and over by four divisions. This gives us the second point to plot.

Step 3: Similar to the previous step, we again move up by three divisions and over by four divisions from the second point. This gives us the third point to plot.

Step 4: As we have already plotted the three points, we join these three points to draw the straight line on the graph.

## Examples

Example 1:

Find the equation of the straight line that has slope of $4$ and contains point having coordinates $(-1, -6)$

Solution:

The slope intercept form of straight line is $y$ = $mx + b$

Plugin the values of $x$ = $-1,\ y$ = $-6$ and $m$ = $4$ in the equation, we get

$-6$ = $4 \times -1 + b$

$-6 + 4$ = $b$

$b$ = $-2$

Therefore, the equation of the straight line would be $y$ = $4x – 2$.
Example 2:

Find the slope intercept form of a straight line having points $(-2, 4)$ and $(1, 2)$.

Solution:

The slope intercept form of straight line is $y$ = $mx + b$

We first find out the slope $(m)$ = $\frac{(2 – 4)}{(1 + 2)}$ = $\frac{-2}{3}$

Plugin the values of $x$ = $-2,\ y$ = $4$ and m = $\frac{-2}{3}$ in the equation, we get

$4$ = $\frac{-2}{3}$ $\times -2 + b$

$4$ = $\frac{4}{3}$ + $b$

$b$ = $4$ – $\frac{4}{3}$

$b$ = $\frac{8}{3}$

Therefore, the equation of the straight line would be $y$ = $(\frac{-2}{3})$ $x$ + $\frac{8}{3}$ or $3y$ = $-2x + 8$.