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Point Slope Equation of a Line

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We know that a line is a plane figure having only length and no width or thickness. Just like all curves, a line can also be represented by an equation. There are many types of equations that can be used to represent a line. Based on the data that is known, we can find the equation of the line. Basically there are three main aspects to the equation of a line. They are its slope and its intercepts. Like any other curve, a line can have two types of intercepts, the $x$ intercept and the $y$ intercept. However, a straight line can have not more than one of each of the intercepts.

Definition

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An equation such that when it is graphed on a grid results in a straight line is called the equation of the line. A line is a plane figure that has only length and no depth or thickness. It extends indefinitely in two directions. The length of a line is infinite. It cannot be measured. The distance between any two points on a given line can be measured. We call that section of a line between two points, a line segment. A line can be thought of as a set of infinitely many points. Equation of a line is also called a linear equation. It can be of one or two variables. Normally we use $x$ and $y$. $x$ is the independent variable and $y$ is the dependent variable.

Formula

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There are primarily three types of lines: Horizontal, Vertical and Slant (or oblique). We shall look at the formulas for equations of each of these types of lines.
Formula for Equation of a Horizontal Line:

On a coordinate grid, a horizontal line would be parallel to the $x$ axis. It would therefore never intercept the $x$ axis, however it would intercept the $y$ axis. If the horizontal line intercepts the $y$ axis in a point $(0,b)$ then the equation of that line can be given by:

$y$ = $b$

See the example below:

Equation of a Horizontal Line

The above example shows the graph of $y$ = $-2$.
Formula for Equation of a Vertical Line:

In case of a vertical line, the line would be parallel to the $y$ axis. So it would pass through some point on the $x$ axis only. Suppose the line passes through the point $(a,0)$. Then the equation of the vertical line would be: $x$ = $a$

Shown below is an example:

Equation of a Vertical Line

The above graph is a graph of a vertical line having equation:

$x$ = $3$
Formula for Equation of a Slant (oblique) Line:

There are more than one formulas for equation of a line. They are as follows:

1) Slope Intercept Form:

This is the most commonly used form for writing the equation of a line. If the slope of a line is $m$ and the $y$ intercept of the line is $b$, then the equation of the line can be given by:

$y$ = $mx + b$

The slope m can be defined as:

$slope$ = $\frac{rise}{run}$

2) Point Slope Form:

If the slope of the line $m$ and a point on the line $(x_1,\ y_1)$ is known to us then we can use the point slope formula to write the equation of the line:

$y - y_1$ = $m(x - x_1)$

3) Intercept Formula:

If we know both the $x$ and the $y$ intercepts of the line, then we can use this formula to write the equation of the line. If the $x$ intercept is $(a, 0)$ and the $y$ intercept is $(0, b)$ then the equation of the line would be:

$\frac{x}{a}$
+ $\frac{y}{b}$ = $1$

4) Standard Form:

The standard form of equation of a line is 
$ax + by + c$ = $0$

Here $a$ and $b$ are respectively the coefficients of $x$ and $y$ and $c$ is the constant.

Examples

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Example 1: 

Write the equation of a line passing through the point $(-4, 3)$ and having a slope of $\frac{-2}{5}$. Then convert  it from point-slope form to slope-intercept form.

Solution:

For this problem our $(x_1,\ y_1)$ = $(-4, 3)$ and the slope $m$ = $\frac{-2}{5}$. The point-slope formula is:

$y - y_1$ = $m(x - x_1)$

Substituting the values we have:

$y - 3$ = $\frac{-2}{5}$ $(x - (-4))$

Simplifying that we have:

$y - 3$ = $\frac{-2}{5}$ $(x + 4)$

This is the equation of the line in point-slope form. Now we shall convert it to slope-intercept form. For that we need to simplify the equation further and solve for $y$.

$y - 3$ = $\frac{-2}{5}$ $x -$  $\frac{8}{5}$

Adding $3$ to both the sides we have:

$y$ = $\frac{-2}{5}$ $x -$ $\frac{8}{5}$ $+ 3$

Simplifying that we have:

$y$ = $\frac{-2}{5}$ $x +$ $\frac{7}{5}$

This is the equation in slope-intercept form. The y intercept as we can see from the equation is $\frac{7}{5}$.
Example 2: 

Write the equation of a line passing through origin and having slope $2$ in point slope form and then convert it to slope intercept form.

Solution:

From the question, point is $(x_1,\ y_1)$ = $(0,\ 0)$ and slope is $m$ = $2$. So the equation of the line in point slope form would be:

$y - 0$ = $2(x - 0)$

Now to convert it to slope intercept form.

$y$ = $2x - 0$

$y$ = $2x\ \leftarrow$ Answer