Coordinate Geometry: Question 1

Worked solution

Step 1: Recall the gradient formula

For two points $(x_1, y_1)$ and $(x_2, y_2)$, the gradient of the line through them is

$m = \frac{y_2 - y_1}{x_2 - x_1}.$

Step 2: Substitute the coordinates

Take $C(-4, 1)$ as $(x_1, y_1)$ and $D(2, -7)$ as $(x_2, y_2)$:

$m = \frac{-7 - 1}{2 - (-4)} = \frac{-8}{6} = -\frac{4}{3}.$

Checking the arithmetic a second time: the change in $y$ is $-7 - 1 = -8$, and the change in $x$ is $2 - (-4) = 2 + 4 = 6$, so $m = -8 \div 6 = -\dfrac{4}{3}$, confirming the same result.

Step 3: Sanity check

Moving from $C$ to $D$, $x$ increases (from $-4$ to $2$) while $y$ decreases (from $1$ to $-7$), so the line falls from left to right. The gradient must be negative. This is consistent with $m = -\dfrac{4}{3}$.

Why the other options are wrong

• B $\left(\dfrac{4}{3}\right)$: correct size, but the negative sign has been lost. This would describe a line that rises from $C$ to $D$, which contradicts the coordinates.
• C $\left(-\dfrac{3}{4}\right)$: the gradient fraction has been inverted, using $\dfrac{\Delta x}{\Delta y}$ instead of $\dfrac{\Delta y}{\Delta x}$.
• D $\left(\dfrac{3}{4}\right)$: both the sign error and the inversion error have been made together.

Final answer

$\boxed{m = -\frac{4}{3}} \quad \text{(Option A)}$