Equation of a straight line

There are 3 forms for the equation of a straight line.

\(y=mx+c\)

This is the most familiar form, where \(m\) is the gradient and \(c\) is the y-intercept. You will usually have to calculate the \(y\)-intercept once you know the gradient and a point the line goes through.

\(y-y_1=m(x-x_1)\)

This is the most useful form, where \(m\) is the gradient and \(x_1\) and \(y_1\) are the co-ordinates of a point that you know the line goes through. This is most useful because it does not require calculation of the \(y\)-intercept.

\(ax+by+c=0\)

This is the least useful form, where \(a\), \(b\) and \(c\) are integers. However, sometimes a question will ask you to give the equation of a line in this form, and you can do so by finding the equation of the line in any of the other 2 forms, then rearranging the equation to get this form.

Calculation of the gradient

If you have two points \((x_1,y_1)\) and \((x_2,y_2)\), then the gradient (\(m\)) of the line that goes through these two points can be calculated by the difference of the \(y\)-coordinates divided by the difference of the \(x\)-coordinates:

\(m=\dfrac{y_2-y_1}{x_2-x_1}\)

The order in which you do the subtraction does not matter, as long as you are consistent.

Parallel and perpendicular lines

For two lines with gradients \(m_1\) and \(m_2\),

• if the lines are parallel, then the gradients are equal, i.e. \(m_1=m_2\);
• if the lines are perpendicular, then the gradients are negative reciprocals of each other, i.e. \(m_1=\dfrac{-1}{m_2}\)

Distance between two points

If you have two points \((x_1,y_1)\) and \((x_2,y_2)\), then the distance (\(d\)) between these two points can be calculated by using Pythagoras' Theorem:

\(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

As the calculation involves squaring the difference of the coordinates, the order in which you subtract does not matter.

Midpoint of a line segment

If you have a line segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\), the midpoint \(M\) can be calculated by finding the average of both the \(x\) and \(y\) coordinates:

\(M=\Big(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\Big)\)

Point of intersection between two lines

The point of intersection between two lines can be found by solving simultaneous equations.

Modelling using straight lines

Straight line graphs can be used for modelling in a variety of contexts, such as the line for converting degrees Celsius to degrees Fahrenheit, distance against time for constant speed, etc.