Finding the equation of a straight line

This method turns geometric information into the algebraic equation of a straight line, usually in gradient-intercept form $y = mx + c$ or the general form $ax + by + c = 0$. The information can be two points that lie on the line, a single point together with the gradient, or a point together with a line that the new line is parallel or perpendicular to.

It answers questions such as 'find the equation of the line joining $A$ and $B$', 'find the equation of the line through $P$ with gradient $2$', or 'find the equation of the perpendicular bisector of $AB$'. Once you have the equation you can find its axis intercepts, test whether a point lies on it, or find where it meets another line, so it is one of the most reused skills in the whole Coordinate Geometry chapter.

Reach for this method whenever a question asks you to 'find the equation of the straight line' and gives enough to fix both its direction (a gradient) and its position (one point it passes through). The common signals are: two named points on the line, a single point plus a stated gradient, or a point plus a parallel or perpendicular condition such as 'parallel to $y = 3x + 1$' or 'perpendicular to $AB$'.

If only a gradient is given with no point, the line is not yet pinned down, search the question for the point. When a parallel or perpendicular line is mentioned, first read off that line's gradient, then convert it: keep the same value for a parallel line, or take the negative reciprocal $m_2 = -\frac{1}{m_1}$ for a perpendicular line, before you substitute a point.

1. 1

   Identify what you are given

   Two points, or one point and a gradient, or one point and a parallel/perpendicular line.

2. 2

   Find the gradient

   From two points use $m = \frac{y_2 - y_1}{x_2 - x_1}$. For a parallel line use the same gradient; for a perpendicular line use $m_2 = -\frac{1}{m_1}$.

3. 3

   Choose one point

   Pick a point $(x_1, y_1)$ that the line passes through, either one works if you have two.

4. 4

   Substitute into point-gradient form

   Write $y - y_1 = m(x - x_1)$ using your gradient and chosen point.

5. 5

   Expand and rearrange

   Multiply out the bracket and collect terms into $y = mx + c$, or the general form $ax + by + c = 0$ if the question asks for it.

6. 6

   Check with the other point

   Substitute a second known point into your equation; both sides should agree.

• Subtracting the coordinates in a different order on the top and bottom of the gradient, for example $\frac{y_2 - y_1}{x_1 - x_2}$, which reverses the sign.
• Getting the perpendicular gradient wrong: the negative reciprocal of $3$ is $-\frac{1}{3}$, not $-3$ or $\frac{1}{3}$.
• Forgetting to expand the bracket in $y - y_1 = m(x - x_1)$, or dropping the sign when the coordinate is negative.
• Leaving the answer as $y - y_1 = m(x - x_1)$ when the question asked for $y = mx + c$ or the general form.
• Swapping which point is $(x_1, y_1)$ partway through, the gradient is the same either way, but you must stay consistent.

The step that costs the most marks is the gradient sign, students subtract $x$ one way and $y$ the other, and the whole line tilts the wrong way. In one-to-one lessons our teachers watch you set up $m = \frac{y_2 - y_1}{x_2 - x_1}$ and insist you label $(x_1, y_1)$ and $(x_2, y_2)$ before any numbers go in, so the subtraction stays consistent top and bottom.

For perpendicular questions we pause on the negative reciprocal until it is automatic. Because SPM Add Math uses analytic marking, we also show you how a clean gradient line earns method marks even if a later arithmetic slip changes $c$.

Every teacher on spmaddmath.com.my is experienced. Lessons are online and taught in English, at your own pace.