Equations of Straight Lines

• $y=mx+c$ is useful when the gradient $m$ and the y-intercept $c$ are known. The y-intercept is the y-value where the line crosses the y-axis. This form cannot represent a vertical line.
• $y-y_1=m(x-x_1)$ is the point-gradient form. Use it when one point and the gradient are known.
• $Ax+By+C=0$ is a general linear form, where $A$ and $B$ are not both zero. If $B\ne0$, isolate $y$ to see that the gradient is $-A/B$. If $B=0$, the line is vertical.

To find an x-intercept, set $y=0$; to find a y-intercept, set $x=0$. For $2x+3y=6$, this gives $(3,0)$ and $(0,2)$ respectively.

For $x\ne x_1$, multiply the gradient equation by $x-x_1$:

$\dfrac{y-y_1}{x-x_1}=m\quad\Longrightarrow\quad y-y_1=m(x-x_1)$

The resulting line equation also includes the original point.

1. Find the required gradient: use the given gradient, calculate it from two points, copy it from a parallel line, or take the negative reciprocal of a perpendicular line.
2. Choose one known point on the required line.
3. Substitute into $y-y_1=m(x-x_1)$. If the required line is vertical, write $x=\text{constant}$ instead.
4. Rearrange to the requested form. Keep exact fractions during the working.

The forms $x+2y=2$, $y=-\frac12x+1$ and $2x+4y=4$ describe the same line. For a “show that” question, finish in the exact printed form rather than with rounded decimals.

At an intersection, one coordinate pair satisfies both line equations. Solve the equations simultaneously by substitution or elimination, then substitute back to find the second coordinate.

Parallel distinct lines have no intersection. Coincident lines are the same line and have infinitely many common points. In either case, solving the equations does not produce one unique coordinate pair.