Point-Slope Form: Writing a Line from One Point

Substitute straight into $y - y_1 = m(x - x_1)$.

$y - 3 = 4(x - 1)$

With $m = 4$, $x_1 = 1$, $y_1 = 3$: $y - 3 = 4(x - 1)$.

In slope-intercept form that is $y = 4x - 1$. Check at $x = 1$: $y = 3$ ✓

$y - 2 = -1(x - 5)$, which is $y = -x + 7$

$y - 2 = -(x - 5) = -x + 5$, so $y = -x + 7$.

Check at $x = 5$: $y = 2$ ✓

Subtracting $-4$ becomes adding $4$.

$y - 1 = 2(x + 4)$, which is $y = 2x + 9$

$y - 1 = 2(x - (-4)) = 2(x + 4) = 2x + 8$, so $y = 2x + 9$.

Check at $x = -4$: $y = -8 + 9 = 1$ ✓

$y + 3 = \tfrac{1}{2}(x - 4)$, which is $y = \tfrac{1}{2}x - 5$

$y - (-3) = \tfrac{1}{2}(x - 4)$, so $y + 3 = \tfrac{1}{2}x - 2$ and $y = \tfrac{1}{2}x - 5$.

Check at $x = 4$: $y = 2 - 5 = -3$ ✓

$y = 3x + 13$

Distribute: $y - 7 = 3x + 6$. Add $7$: $y = 3x + 13$.

Look at what happens when $x_1$ is $0$.

$y - 8 = -3(x - 0)$, which becomes $y = -3x + 8$ directly.

Since $x_1 = 0$, the bracket reduces to $x$, and adding $8$ gives $y = -3x + 8$.

When the point you are given is the $y$-intercept, point-slope collapses straight into slope-intercept form — the two are the same idea seen from different starting information.

Substitute both coordinates and see whether the two sides agree.

Yes.

Left: $12 - 2 = 10$. Right: $5(3 - 1) = 10$. The sides match, so the point is on the line.

Losing volume makes the slope negative. The starting volume is the value at $x = 0$.

$y = -600x + 5400$; it started with $5400$ litres.

Slope $-600$ through $(2, 4200)$: $y - 4200 = -600(x - 2)$.

Expanding: $y - 4200 = -600x + 1200$, so $y = -600x + 5400$.

At $x = 0$ the volume is $5400$ litres. Check at $x = 2$: $-1200 + 5400 = 4200$ ✓