Gradient, slope-intercept, point-slope and standard forms

The gradient formula is $m=\frac{y_2-y_1}{x_2-x_1}$ where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line. Here, $(x_1,y_1)=(1,3)$ and $(x_2,y_2)=(5,11)$.

Substituting into the formula: $m=\frac{11-3}{5-1}=\frac{8}{4}$ Perform the subtraction in both numerator and denominator.

$m=\frac{8}{4}=2$ The gradient is $m=2$, which is positive, confirming the line slopes upward from left to right.

We need the gradient between $(1,3)$ and $(5,11)$. The rise is $11-3=8$ and the run is $5-1=4$.

A negative gradient would mean the line slopes downward. Since $y$ increases from 3 to 11 as $x$ increases, the gradient must be positive. Eliminate $m=-2$.

$m=4$ would mean the rise is 4 times the run. The rise is 8 and the run is 4, so the ratio is $\frac{8}{4}=2$, not 4. Eliminate $m=4$.

$m=\frac{1}{2}$ would mean the rise is half the run. The rise is 8 and the run is 4, giving ratio 2, not $\frac{1}{2}$. Eliminate $m=\frac{1}{2}$.

The only remaining option is $m=2$, which matches $\frac{8}{4}=2$. This is correct.

The slope-intercept form is $y=mx+c$, where $m$ is the gradient and $c$ is the y-intercept. We need to match each part of $y=-4x+7$ to this template.

Comparing $y=-4x+7$ with $y=mx+c$: the coefficient of $x$ is $-4$, so $m=-4$. The constant term is $+7$, so $c=7$.

The gradient is $m=-4$ (negative means the line slopes downward from left to right) and the y-intercept is $c=7$ (the line crosses the y-axis at the point $(0,7)$).

In $y=mx+c$, the number multiplying $x$ is the gradient and the constant added at the end is the y-intercept. In $y=-4x+7$, the number with $x$ is $-4$ and the constant is $+7$.

This option has the wrong signs for both values. The coefficient of $x$ is $-4$, not $+4$, and the constant is $+7$, not $-7$. Eliminate this option.

This option has the values swapped. The number 7 is the constant term (y-intercept), not the gradient. Eliminate this option.

The gradient $-4$ is correct, but the y-intercept is wrong. The equation has $+7$, not $-7$. Eliminate this option.

The only remaining option, 'Gradient $=-4$, y-intercept $=7$', correctly identifies both values from $y=-4x+7$.