y-b=m(x-a) (SQA National 5 Maths): Revision Note

Determining the equation of a straight line

What is the equation of a straight line?

• The general equation of a straight line is y = mx + c  where

  • m  is the gradient

  • c  is the y-intercept

    • The value where the line cuts the y-axis

• y  = 5x  + 2  is a straight line with

  • gradient 5

  • y-intercept 2

• y  = 3 - 4x  is a straight line with

  • gradient -4

  • y-intercept 3

How do I find the equation of a straight line using y-b=m(x-a)?

• To find the equation of a line you need to know

  • The coordinates of a point on the line

  • The gradient of the line

    • If you know two points on the line, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$

    • You can find the gradient using the formula $\mathrm{gradient}=\frac{y_2-y_1}{x_2-x_1}$

• If the line has gradient m and goes through the point (a, b)

  • then an equation of the line is y - b = m(x - a)

  • E.g. the line with gradient 3 through point (2, 5)

    • $m=3$, $a=2$, $b=5$

    • An equation of the line is $y-5=3\left(x-2\right)$

• An equation in this form can be rearranged into y = mx + c form if required:

$\begin{array}{rcl}y-5& =& 3\left(x-2\right)\\ y-5& =& 3x-6\\ y& =& 3x-6+5\\ y& =& 3x-1\end{array}$

How do I find the equation of a straight line using y=mx+c?

• You can also use y = mx + c to find the equation of a line, if you know the gradient m and a point on the line (a, b)

• Substitute the values of m and (x, y) = (a, b) into the equation

  • Then solve to find the value of c

• E.g. the line with gradient 3 through point (2, 5)

  • $m=3$, $\left(x,y\right)=\left(2,5\right)$

    • Substituting into $y=mx+c$gives $5=3\left(2\right)+c$

    • So $5=6+c\Rightarrow c=-1$

  • The equation of the line is $y=3x-1$

How do I find the equation of a straight line from a graph?

• Find the gradient m by drawing a triangle and using

  • $\mathrm{gradient}=\frac{\mathrm{rise}}{\mathrm{run}}$

    • Positive for uphill lines, negative for downhill

• Read off the y-intercept c from the graph

  • Where it cuts the y-axis

• Substitute these values into y  = mx  + c 

What if no y-intercept is shown on the graph?

• If you can't read off the y-intercept

  • Find any point on the line

  • Use one of the methods above to find the equation

What are the equations of horizontal and vertical lines?

• A horizontal line has the equation y  = c

  • c  is the y-intercept

• A vertical line has the equation x  = k

  •  k  is the x-intercept

• For example

  • y = 4

  • x = -2