Equation of a Line Through Two Points

Finding a line through two points is a two-step staple: gradient first, then a point. It underpins many longer 0606 questions.

Find the equation of the straight line passing through the points $A(2, 3)$ and $B(4, 7)$. [3]

The working

Step 1, find the gradient $m = \dfrac{y_2 - y_1}{x_2 - x_1}$: $m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \quad \text{(M1)}$

Step 2, use one point with $y - y_1 = m(x - x_1)$, taking $A(2, 3)$: $y - 3 = 2(x - 2) \quad \text{(M1)}$

Step 3, rearrange: $y - 3 = 2x - 4 \;\Rightarrow\; y = 2x - 1 \quad \text{(A1)}$

A quick check: does $B(4, 7)$ satisfy it? $2(4) - 1 = 7$ ✓.

Where the marks are won and lost

• Compute the gradient with a consistent subtraction order: $\frac{7-3}{4-2}$, both differences the same way round. Flipping one gives the wrong sign.
• Either point works in $y - y_1 = m(x - x_1)$; the resulting line is identical. Substituting a point’s coordinates in the wrong slots is the usual slip.
• Check with the other point, it should satisfy the equation.

Common mistakes

• Inconsistent order in the gradient (e.g. $\frac{7-3}{2-4}$), giving the wrong sign.
• Mixing up which coordinate is $x_1$ and $y_1$ in the point-slope form.
• Arithmetic slips rearranging to $y = mx + c$.

Full method: Equation of a Line notes. Topic home: Straight-Line Graphs pillar.