y = mx + c (AQA GCSE Maths): Revision Notes

The equation of a straight line: y = mx + c

Understanding the standard form

The equation $y = mx + c$ is the most common way to write straight-line equations, and it's particularly useful for GCSE exams. This format is called the standard form or slope-intercept form.

In this equation:

• m represents the gradient (or slope) of the line
• c represents the y-intercept (the point where the line crosses the y-axis)

Rearranging equations into standard form

Many straight-line equations don't initially appear in the $y = mx + c$ format. The first step is often to rearrange them into this standard form. This involves using algebraic manipulation to isolate y on one side of the equation.

Finding the equation from a graph

When you're given a graph of a straight line, you can determine its equation by identifying two key features:

1. The y-intercept (c): Look for where the line crosses the y-axis
2. The gradient (m): Calculate how much the line rises or falls for each unit it moves horizontally

The gradient is calculated using the formula:

$\text{gradient} = \frac{\text{change in y}}{\text{change in x}}$

Once you have both values, simply substitute them into the $y = mx + c$ format to get your equation.

Finding the equation using two points

If you're given two points that lie on a straight line, you can find the equation using this method:

1. Calculate the gradient: Use the formula $m = \frac{\text{change in y}}{\text{change in x}} = \frac{y\_2 - y\_1}{x\_2 - x\_1}$
2. Substitute into the general form: Use $y = mx + c$ and substitute one of the points along with your calculated gradient
3. Solve for c: Rearrange to find the y-intercept value
4. Write the final equation: Substitute both $m$ and $c$ back into $y = mx + c$

Important reminders

Converting between different forms of linear equations is a common exam requirement. You might be asked to express your answer in forms like $ax + by + c = 0$, but the $y = mx + c$ method is usually the most straightforward approach to start with.

Practice identifying the gradient and y-intercept from both graphs and equations, as this skill appears frequently in GCSE mathematics papers.