Equation of a straight line: $$y=mx+c$$​​

In a nutshell

The equation $$y=mx+c$$ gives a straight line on a coordinate grid, where $$m$$ and $$c$$are constants. It is the equation for almost any straight line, the exception being vertical lines, which have equations of the form $$x=d$$ where $$d$$ is a constant (the $$x$$-intercept).

The components of the equation $$y=mx+c$$

$$m$$​ is the value of the gradient of the straight line and $$c$$ is the $$y$$-intercept. $$x$$ and $$y$$correspond to coordinates of points on the line. For any point $$(x,y)$$ on the line, multiplying the $$x$$-coordinate by $$m$$ and adding $$c$$, gives the $$y$$-coordinate. If this doesn't work, then the point you are using is not actually on the line.

Example 1

Consider the equation of the line below. Does the point $$(-1,-5)$$ sit on this line?

$$y=3x-2$$​​

Insert the $$x$$-coordinate $$-1$$ into the equation of the line to see if the equation then gives the corresponding $$y$$-coordinate:

​$$y=3x-2=3(-1)-2=-3-2=-5$$​​

This is the correct $$y$$-value, therefore:

$$(-1,-5)$$ does sit on the line $$y=3x-2$$

Example 2

Consider the line with equation below. Does the point $$(4,9)$$ sit on this line?

$$y=3x-2$$​​

Insert the $$x$$-coordinate into the equation of the line. If it gives the $$y$$-coordinate, then the point is on the line, if it doesn't, then it is not.

​$$y=3x-2=3(4)-2=12-2=10$$​​

This is not $$9$$, so:  

$$(4,9)$$ is not on the line $$y=3x-2$$​

Note: Point $$(4,10)$$ is, however, on the line.

Different types of lines

​​​​Diagonal lines

Diagonal lines have the equation $$y=mx+c$$ where $$m$$ is the gradient and $$c$$ is the $$y$$-intercept.

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​​Horizontal lines

Horizontal lines have a gradient of $$0$$, so such a line is just $$y=c$$. This is still technically in the form $$y=mx+c$$, but $$m$$ is equal to $$0$$.​

Vertical lines

Vertical lines don't use the $$y=mx+c$$ equation. You cannot give $$m$$ a value for a vertical line since it is essentially infinity; there is also no value for $$c$$. Instead, a vertical line has the equation $$x=d$$ where $$d$$ is the $$x$$-intercept.

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Example 3

By looking at the following equations of lines, decide which are horizontal, which are vertical and which are diagonal.

Diagonal lines have equations of the form $$y=mx+c$$ where $$m$$ is not zero. Horizontal lines have equations of the form $$y=c$$ and vertical lines have equations of the form $$x=d$$. Importantly, $$c$$ and $$d$$ are constants, so do not include any $$x$$ or $$y$$terms. Thus:

Diagonal lines: b, c and e​

Horizontal lines: a​

Vertical lines: d

Example 4

For each of the diagonal lines in the example above, what is the gradient?

The gradient is $$m$$, which is the number that sits before the $$x$$. The gradients are as follows:

b: $$\underline4$$

c: $$\underline{-7}$$

e: $$\underline{1}$$​

Note: In line e, you don't need to write the $$1$$ ahead of the $$x$$​ - it is implied.