$m$ is the gradient (steepness/direction), and $c$ is the $y$-intercept — the value of $y$ where the line crosses the $y$-axis (i.e. where $x=0$). Once you know $m$ and $c$, you know the entire line.

Read $c$ straight off the graph (where it crosses the $y$-axis), then find $m$ using rise over run between two clearly-plotted points (see E3.3). Substitute both into $y = mx+c$.

This combines two skills you already have: use the gradient formula from E3.3 to calculate $m$ from the two points, then substitute $m$ and one of the points into $y=mx+c$ and solve for $c$. This method works even when there's nothing to read off a diagram.

Lines are sometimes given as $ax + by = c$ or simply $x = k$. For $5x + 4y = 8$:

A line of the form $x = k$ is a vertical line — it has no gradient in the $y=mx+c$ sense and cannot be written that way at all.

Cambridge expects fractions cancelled and no unnecessary decimals — e.g. write $y = -\tfrac{5}{4}x + 2$ rather than $y = -1.25x + 2$ unless a decimal is specifically requested.

Substitute both of the original coordinates back into your equation — if a point genuinely lies on the line, its $x$ and $y$ values must satisfy $y = mx + c$ exactly. This costs seconds but catches almost every arithmetic slip, particularly sign errors made while solving for $c$, and is well worth doing on any "find the equation" question worth more than one mark.

Find the gradient and $y$-intercept of the line with equation $5x + 4y = 8$.

1. Rearrange to make $y$ the subject: $4y = 8 - 5x$.
2. Divide every term by $4$: $y = 2 - \tfrac{5}{4}x$, i.e. $y = -\tfrac{5}{4}x + 2$.
3. Compare with $y = mx+c$: gradient $m = -\tfrac{5}{4}$, $y$-intercept $c = 2$.

Find the equation of the line passing through $A(2, 1)$ and $B(6, 9)$.

1. Find the gradient: $m = \dfrac{9-1}{6-2} = \dfrac{8}{4} = 2$.
2. Substitute the gradient and one point, say $A(2,1)$, into $y = mx + c$: $1 = 2(2) + c \implies 1 = 4 + c \implies c = -3$.
3. The equation of the line is $y = 2x - 3$.
4. Check with the other point: $B(6,9)$ gives $2(6)-3 = 9$ ✓, confirming the equation is correct.