7.1 Straight Line Equations

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📖 Main Content

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Form 1: Slope-Intercept Form

$y = mx + c$

This is the most common form. Here is what each letter means:

• $y$ and $x$ are the coordinates of any point on the line
• $m$ is the gradient — this tells you how steep the line is
• $c$ is the y-intercept — this is the value of $y$ where the line crosses the y-axis (the vertical axis)

Example: In the equation $y = 3x + 5$:

• The gradient is $m = 3$ (the line goes up 3 units for every 1 unit across)
• The y-intercept is $c = 5$ (the line crosses the y-axis at the point $(0, 5)$)

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Form 2: Point-Gradient Form

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

This form is very useful when you know the gradient of a line and one point that the line passes through.

Here is what each letter means:

• $m$ is the gradient of the line
• $(x_1, y_1)$ is the known point on the line (a specific coordinate you are given)
• $(x, y)$ is any general point on the line

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2. How to Find the Gradient

Before you can write the equation of a line, you often need to find its gradient first.

The gradient formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

In plain English: gradient = change in $y$ ÷ change in $x$

• If the line goes uphill (left to right), the gradient is positive
• If the line goes downhill (left to right), the gradient is negative
• If the line is horizontal (flat), the gradient is zero