# Equations of Straight Lines

GCSELevel 4-5Level 6-7Cambridge iGCSE

## Equations of Straight Lines

Straight line equations are a rule for generating straight lines on an $x$$y$ plane.

Every equation for a straight line takes the same form:

$y=mx+c$

Where $y$ and $x$ are the variables and $m$ and $c$ are constants that define the straight line.

## The Equation

$y=mx+c$

$x$ & $y$ are the Cartesian coordinates. These are graphed into an $x$-axis and a $y$-axis, that shows how $y$ varies as the input of $x$ changes. If $y=4$ when $x=3$ we would write this coordinate as $(3, 4)$ (the $x$ coordinate always goes first).

$m$ is the gradient of the line. Which can be found from two points on the line. If $(x_1, y_1)$ and $(x_2, y_2)$ are points on the straight line then,

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

Where $\Delta x$ means the change in $x$ (in this case between $x_2$ and $x_1$).

$c$  is known as the $y$-intercept. It is the value of $y$ when $x=0$. So the point $(0,c)$ is on the line, and is where the line crosses the $y$-axis. $c$ can be found if you have the gradient $m$ and a point on the line $(x_1, y_1)$.

$c =y_1 - mx_1$

Level 4-5GCSECambridge iGCSE

## Graphical Interpretation

When a straight line has been graphed the equation can be deduced. Where the line crosses the $y$-axis is the $y$-intercept, $c$. You can then pick two points on the line and find the difference in $y$ and the difference in $x$ between the points. When the line is drawn on a graph, you can pick the points for ease of calculation.

So if the line passes through $(x_1,y_1)$ and $(x_2,y_2)$ then $m = \dfrac{y_1-y_2}{x_1-x_2}$

You may see on the graph, that for every $1$ unit $x$ increases, $y$ increases by $m$ units.

Remember if $y$ decreases as $x$ increases then the gradient is going to be negative.

Level 4-5GCSECambridge iGCSE

## Example 1: Deriving the Equation From a Graph

From the graph given, we want to derive an equation for the straight line.

[3 marks]

The graph crosses the $y$-axis at the point $(0,4)$, so the $y$-intercept, $c$, is equal to $4$. We must now find another point to find the gradient.

Looking at another point, let’s choose $(3,6)$ we can find the gradient.
$\Delta y = 6-4=2$
$\Delta x = 3-0$ so,

$m=\dfrac{2}{3}$

Then the equation for the straight line is

$y=\dfrac{2}{3} x+4$

Level 4-5GCSECambridge iGCSE

## Example 2: Straight Lines Between Two Points

Any two points on the coordinate plane can be connected by a straight line. So we must be able to find an equation for this straight line. So lets find the equation to the line that passes through $(1, 4)$ and $(2, 7)$.

[3 marks]

Firstly we must find the gradient, as we cannot find the $y$-intercept without it.

$\Delta y = 7-4 = 3$
$\Delta x = 2-1 = 1$, so

$m=\dfrac{3}{1}=3$

Now we have $y=3x+c$, so what we can do is substitute one of our points on the graph into the equation to find $c$, we shall choose $(1,4)$ although it doesn’t matter which one.

$y=3x+c \rightarrow 4=3 \times 1 +c$
$4-3=c$
$c=1$

So the equation is $y=3x + 1$

Level 6-7GCSECambridge iGCSE

## Equations of Straight Lines Example Questions

Straight line equations take the form $y=mx+c$

We can clearly see that the red line crosses the $y$-axis at the point $(0,-1)$, so $c=-1$

Taking another point on the line, choose $(3,0)$ we shall find $m$.

$m = \dfrac{\Delta y}{\Delta x} = \dfrac{-1-0}{0-3}= \dfrac{1}{3}$

So the equation for the red line is

$y=\dfrac{1}{3}x-1$

Gold Standard Education

a) We need to put this equation into the form $y=mx +c$

$2x=\dfrac{y}{2}-4$

$4x=y-8$

$y=4x+8$

So the gradient is $8$

b) The point that the line crosses the $x$-axis is when $y=0$, so

$2x=\dfrac{0}{2} -4$

$2x=-4$

$x=-2$

The line crosses the $x$-axis at the point $(-2,0)$

Gold Standard Education

First, we shall find the gradient of the line that passes through $(3,3)$ and $(6,0)$.

$m=\dfrac{\Delta y}{\Delta x}$

$\Delta y = 0-3= -3$
$\Delta x = 6-3=3$

So

$m=\dfrac{-3}{3}=-1$.

To find $c$ we shall substitute in the coordinates $(3,3)$

$3= (-1)\times 3 + c$

$c= 3+3 = 6$

The equation for the straight line is $y=-x+6$