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Equation of a straight line: y = mx + c

Tutor: Toby

# 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 lin$y=3x-2$

##### 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.

### ​​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.

##### Example 3

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

 a) $y=6$​​ b) $y=4x-3$​​ c) $y=-7x$​​ d) $x=0$​​ e) $y=x+2$​​

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: bc 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.

## FAQs - Frequently Asked Questions

### What is the equation of a straight line?

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