Equations of straight lines

In a nutshell

You can find out the equation of a straight line, $y=mx+c$ , using gradients and points. The equation of a straight line can be found by using a pair of coordinates $(x_1,y_1)$ and the gradient $m$ . You can use an equation, $y-y_1=m(x-x_1)$ , to find the exact equation for $y=mx+c$ . You can also use equations to find where lines intersect.

Finding $y=mx+c$ using the gradient and a point

If you know the gradient of a line, $m$ , and you know a point on that line, $(x_1,y_1)$ , then you can work out the equation of that line. All you have to do is use the formula $y-y_1=m(x-x_1)$ . This will allow you find the line's equation.

Procedure

1.	Label the gradient and the coordinates of the point on the line.
2.	Substitute into the equation $y-y_1=m(x-x_1)$ .
3.	Solve for $y=mx+c$ .

Example 1

A line has a point $(3,4)$ and a gradient of $2$ . Find the equation of the line in the form of $y=mx+c$ .

Label the gradient and the coordinates.

$m=$ $2$ , $x_1=$ $3$ , $y_1=4$

Substitute $y_1$ , $m$ and $x_1$ in the equation $y-y_1=m(x-x_1)$ .

$y-4=2(x-3)$

Rearrange to give $y=mx+c$ .

$\begin{aligned}y-4 &= 2x-6\\&\underline{y =2x-2}\\\end{aligned}$

Finding $y=mx+c$ using two points

To find the equation of a line using two points $(x_1, y_1)$ and $(x_2,y_2)$ , first you find the gradient $m$ using the equation from the previous lesson, $m=\dfrac{y_2-y_1}{x_2-x_1}$ . Once you find out $m$ , you can then take the $(x_1, y_1)$ coordinates and solve as above.

Procedure

1.	Label the coordinates $(x_1,y_1)$ and $(x_2,y_2)$ .
2.	Find out the gradient $m$ .
3.	Rearrange for $y=mx+c$ .

Example 2

A line intersects two points, $P_1$  with coordinates $(-2, 5)$ and $P_2$ with coordinates $(-6,1)$ . Find the equation of the line in the format $y=mx+c$ .

Label both points.

$x_1=-2, y_1=5, x_2=-6,y_2=1$

Find the gradient $m$ .

$\begin{aligned}m&=\dfrac{y_2-y_1}{x_2-x_1}\\ \\m&=\dfrac{1-5}{-6-(-2)}\\ \\m&=\dfrac{-4}{-4}\\ \\m&=1\\\end{aligned}$

Solve for $y=mx+c$ .

$\begin{aligned}y-y_1&=m(x-x_1)\\y-5&=1(x-(-2))\\y&= x +2+5\\&\underline{y=x+7}\\\end{aligned}$ ,

Find the intersection between two lines

Finally, sometimes you will be given two line equations and asked to find a point where those lines intersect. This can be solved like simultaneous equations, where you substitute $x$ or $y$ in one equation with an answer from the other. This works mathematically since the point where those lines intersect means that the $x$ and $y$ values in both equations must be equal.

Example 3

The lines $y=2x+5$ and $5x-2y+12=0$ intersect at the point $A$ . Find the coordinates of $A$ .

Substitute $y$ in one equation.

$\begin{aligned}5x-2y+12&= 0\\2x+5&=y\\5x-2(2x+5)+12&=0\\\end{aligned}$

Solve for $x$ 

$\begin{aligned}5x-4x-10+12&=0\\x+2&=0\\x&=-2\\\end{aligned}$

Use this $x$ value to solve for the corresponding $y$ value to find point $A$ .

$\begin{aligned}y&=2x+5\\y&=2(-2)+5\\y&=-4+5\\y&=1\\&\underline{\therefore\space A =(-2,1)}\\\end{aligned}$