Equation of a Straight Line (AQA A-Level Mathematics): Revision Notes

Example:

Given the slope $m = 2$ and $y$-intercept $c = -3$, the equation of the line is:

$y = 2x - 3$

This line has a slope of $2$ and crosses the $y$-axis at $(0, -3)$.

Example:

Suppose you know a line passes through the point $(3, 4)$ and has a slope of $m = -2$. The equation of the line is:

$y - 4 = -2(x - 3)$

Simplifying this:

$y = -2x + 6 + 4$

$y = -2x + 10$

Example:

The line $y = 2x - 3$ can be written in general form by rearranging the terms:

$2x - y - 3 = 0$

Example:

Given points $(1, 2)$ and $(4, 8)$, the equation of the line is:

$y - 2 = \frac{8 - 2}{4 - 1}(x - 1)$

$y - 2 = 2(x - 1)$

$y = 2x - 2 + 2$

$y = 2x$

Example:

• Deduce from rearrangement the gradient and y-intercept of $6x + 5y = 3$: $5y = 3 - 6x$ $y = \frac{3}{5} - \frac{6}{5}x$
• Gradient $m = -\frac{6}{5}$
• $y$-intercept $c = \frac{3}{5}$

Example:

• Find the equation of the line that passes through $(2, 7)$ and has a gradient $m = 4$: $y - 7 = 4(x - 2)$ $y - 7 = 4x - 8$

$y = 4x - 1$

• The equation of the line in slope-intercept form is $\boxed {y = 4x - 1}$.

Problem: Find the equation of the line that passes through $(4, 7)$ and $(9, -12)$ in the form $ax + by + c = 0$, where $a, b, c \in \mathbb{Z}$.

Solution:

1. Calculate the Gradient: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-12 - 7}{9 - 14} = \frac{-19}{-5} = \frac{19}{5}$
2. Use the Point-Slope Form: $y - y_1 = m(x - x_1)$ Using the point $(4, 7)$: $y - 7 = \frac{19}{5}(x - 14)$
3. Clear the Fraction: Multiply both sides by $5$: $5(y - 7) = 19(x - 14)$ $5y - 35 = 19x - 266$
4. Rearrange to Standard Form: $5y - 35 = 19x - 266$ $5y - 19x + 231 = 0$
5. General Form: $-19x + 5y + 231 = 0$

Notes:

• We could use $(9, -12)$ instead of $(4, 7)$, but using positive numbers simplifies calculations.
• Multiplying both sides by 5 ensures all coefficients are integers.
• Any integer multiple of the equation is also acceptable, e.g., $-1900x + 500y + 23100 = 0$, but the simplest form is preferable.

Summary:

• Slope-Intercept Form: $y = mx + c$
• Point-Slope Form: $y - y_1 = m(x - x_1)$
• General Form: $Ax + By + C = 0$
• Two-Point Form: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$