$$y=mx+c$$​

​​In a nutshell

$$y=mx+c$$ is the equation for a straight line, where $$m$$ is the gradient of the line and $$c$$ is the $$y$$-axis intercept. $$m$$​ tells you how steep a line is by tracking the change in $$y$$​ over the change in $$x$$​. $$c$$​ tells you where the line intercepts the $$y$$-axis. This is a handy equation because it allows you to learn a lot about a line with only a pair of coordinates. Once you understand it intuitively, you can plot out lines without graph paper.

The gradient formula

A gradient tells you how steep a line is between two points $$(x_1,y_1)$$ and $$(x_2, y_2)$$. It's calculated as: 

$$\boxed{m = \dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}}$$

Intuitively, you can see that a greater change in $$y$$​ or $$x$$ would lead to a steeper line on paper. Therefore, a higher $$m$$ value means a greater gradient.

Three lines with the equations $$y=x+4$$, $$y=5x+4$$ and $$y=-5x+4$$.

Once you have the coordinates of a line, you can work out the gradient easily.

Procedure

Example 1

Find the gradient for a line between points $$(2,3)$$ and $$(6,7)$$.

Assign values for $$(x_1,y_1)$$ and $$(x_2,y_2)$$.

 $$x_1 = 2$$, $$y_1 = 3$$, $$x_2= 6$$ and $$y_2=7$$

Substitute into the formula for $$m$$ and solve.

$$\begin{aligned}m&=\dfrac{y_2-y_1}{x_2-x_1}\\\\m&= \dfrac{7-3}{6-2}\\\\m&= \dfrac{4}{4}\\\end{aligned}$$​​

$$\underline{m=1}$$​​

Working backwards

You can works backwards if you are given just the gradient of a line and some coordinates. This can let you find where a line terminates.

Example 2

A line has coordinates $$(4,8)$$ and $$(-1, a)$$. The gradient of the line is $$-2.5$$​. What is the value of coordinate a?

Assign values to $$(x_1,y_1)$$ and $$(x_2,y_2)$$

​$$\begin{aligned}(4,8)&(-1,a)\\(x_1,y_1)&(x_2, y_2)\\x_1 = 4, x_2 &= -1, y_1=8, y_2=a\\m&=-2.5\\\end{aligned}$$​​

Substitute into the formula for $$m$$

​​

 $$\begin{aligned}m&=\dfrac{y_2-y_1}{x_2-x_1}\\\\-2.5 &=\dfrac{a-8}{-1-4}\\\end{aligned}$$

Solve for $$a$$​

$$\begin{aligned}\dfrac{a-8}{-5}&=-2.5\\a&= 12.5 + 8 \\&\underline{a=20.5}\\\end{aligned}$$

The y-intercept, c

​$$c$$ is known as the $$y$$​-intercept. It is the point where the line has an $$x$$​ value of zero. Visually, this is the point where the line hits the $$y$$​-axis, as the $$y$$​-axis lies where $$x=0$$​. To find the intercept from a complete line equation, all you need to do is make $$x=0$$​, i.e, remove the $$m$$​ and $$x$$​ from the equation.

Three lines with equations $$y=x+4$$, $$y=x+2$$ and $$y=x-4$$.

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The $$x$$-intercept is needed less often. It is simply where the line hits the $$x$$​-axis. To find it, you make $$y=0$$​ and solve accordingly.

Example 3

What is the gradient and the y-intercept of line $$3y+ 9x - 12 = 6$$?

Rearrange the equation into the form $$y=mx + c$$ and solve.

​$$\begin{aligned}3y+9x-12&=6\\3y &= -9x +12 + 6\\3y&=-9x + 18\\y&=-3x + 6\\&\underline{m= -3, c= 6}\\\end{aligned}$$​​

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