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Slope Intercept Equation

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The slope intercept form is a form of equation of a line. Out of the many ways of writing an equation of a line, the slope intercept form is one of the most commonly used one. Besides this there are other forms such as point slope form, intercept form, standard form, etc. A given straight line can be represented by an equation in any of these forms. Whether the line is horizontal, vertical or oblique, it can always be represented by any of the above forms of equations of a line. The orientation of a line can be determined by the slope of the line.

Definition

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The slope intercept form of equation of a line comprises of two components: the slope of the line and the $y$ intercept of the line.

Slope: The slope of a line is defined as the ratio of rise over run. The altitude covered by a line per unit horizontal distance is called the slope of the line. If the slope of the line is zero then the equation represents a horizontal line parallel to $x$ axis. 

Horizontal Line Equation 

If the slope of the line is negative, then the orientation of the line would look as follows:

Slope of a Line 

If the slope of line is positive then it would look as follows:

 Slope Intercept Equation

The intercept of a line used in this form of equation is the $y$ intercept. The $y$ intercept is defined as the $y$ co-ordinate of the point where the graph of the line cuts (or intercepts) the $y$ axis. For instance, in the above graph the $y$ intercept is at $(0,\ -1)$, so the $y$ intercept would be $-1$.

Formula

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The formula for slope intercept form of equation of a line is:
$y$ = $mx\ +\ b$

Here, $m$ is the slope of the line and $b$ is the $y$ intercept of the line. The slope $m$ can be calculated using any two points that lie on the line. If any two points on a line are $(x_1,\ y_1)$ and $(x_2,\ y_2)$ then they would look as follows:

Slope Intercept Formula
 
In the above figure note that the rise would be the difference between the y coordinates. Thus, rise = $y_2\ -\ y_1$

The run would be the difference between the x coordinates. So, run = $x_2\ -\ x_1$

So the slope can be calculated as: slope = $m$ = $\frac{(y_2\ -\ y_1)}{(x_2\ -\ x_1)}$

Once we know this slope, we can write the equation of the line using the point slope formula as follows:
$y\ -\ y_1$ = $m(x\ -\ x_1)$

Now, distributing the slope to the $x\ -\ x_1$ we have:
$y\ -\ y_1$ = $mx\ -\ mx_1$

To solve that for $y$, we can add $y_1$ to both the sides, that gives us:
$y$ = $mx\ +\ (y_1\ -\ mx_1)$

This expression: $y_1\ -\ mx_1$ would be the $y$ intercept of the line. Let us replace that with a number $b$. Thus, now we have the formula for equation of line as:
$y$ = $mx\ +\ b$

This is called the slope intercept form of equation of a line.

Slope Intercept Equation of Vertical & Horizontal Lines

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Let us first talk of horizontal lines. Consider the following graph of a horizontal line. 

Slope Intercept Equation of Horizontal Line
 
Note that there are two points marked on that line. From the graph we see that the $y$ co ordinate of both the points would be the same. Let us say that it is equal to $b$. Then the coordinates of the two points would be:
$A(x_1,\ b)$ and $B(x_2,\ b)$

Now, our formula for slope was:

$m$ = $\frac{(y_2\ -\ y_1)}{(x_2\ -\ x_1)}$

Substituting the values of the coordinates of points $A$ and $B$ we have:

$slope$ = $m$ = $\frac{(b\ -\ b)}{(x_2\ -\ x_1)}$

Simplifying that we have:

$slope$ = $m$ = $\frac{0}{(x_2\ -\ x_1)}$ = $0$

Thus we see that the slope of a horizontal line would always be zero. 

Now, from the graph we can see that this horizontal line has a $y$ intercept of $b$. This is because the $y$ coordinates of all the points on the line would be same $b$. Thus using the slope intercept form the equation of a horizontal line would be:

$y$ = $0\ x\ +\ b$

$y$ = $b$

Now let us consider a vertical line. Given below is a graph of a vertical line with two points marked on it.

Slope Intercept Equation of Vertical Line
 
This time note that the $x$ coordinates of the points $A$ and $B$ would be same. Let us say that it is equal to $a$. Then the coordinates of the two points would be: $A(a,\ y_1)$ and $B(a,\ y_2)$. Plugging these into the slope formula we have:

$slope$ = $m$ = $\frac{(y_2\ -\ y_1)}{(x_2\ -\ x_1)}$

$slope$ = $m$ = $\frac{(y_2\ -\ y_1)}{(a\ -\ a)}$

$slope$ = $m$ = $\frac{(y_2\ -\ y_1)}{0}$

$slope$ = $m$ = $not\ defined!$

Division by zero is not defined, therefore the slope of a vertical line is not defined. So what do we do? How do we write the equation of a vertical line? Well the answer is simple, since the slope is not defined, we cannot use the slope intercept form $y$ = $mx\ +\ b$ here. However we can take a hint from the equation of the horizontal line that we did earlier. Since the horizontal line had a $y$ intercept of $b$ the equation of the line became $y$ = $b$, similarly the vertical line has an $x$ intercept of a so the equation of the vertical line would become: $x$ = $a$

Examples

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Example 1: 

Write the equation of a line in slope intercept form that has a slope of $\frac{-2}{3}$ and a $y$ intercept of $5$

Solution:

The formula for slope intercept form is:

$y$ = $mx\ +\ b$

For our problem we are given that:

$slope$ = $m$ = $\frac{-2}{3}$

And that

$y$ intercept = $b$ = $5$

Therefore the equation of the line would be:

$y$ = $\frac{-2}{3}$ $x\ +\ 5$

Above is the answer!
Example 2: 

Write the equation of the line represented by the following graph:

Slope Intercept Equation Example
 
Solution:

From the graph we see that there are two points marked which are:

$(x_1\ y_1)$ = $(2,\ -2)$

$(x_2,\ y_2)$ = $(-1,\ 4)$

Using the formula for slope we have:

$slope$ = $m$ = $\frac{(y_2\ -\ y_1)}{(x_2\ -\ x_1)}$

Substituting the values we have:

$m$ = $\frac{(4-(-2))}{(-1\ -2)}$

Simplifying that we have:

$m$ = $\frac{6}{(-3)}$ = $-2$

Also from the graph we see that the $y$ intercept of the line is at $(0,\ 2)$

So the value of $b$ = $2$. Thus the equation of the line in slope intercept form would be:

$y$ = $-2x\ +\ 2$

Answer!