Solve for y: (y + 397)/(−409 + 397) = (x − 362)/(366 − 362).

Step-by-step solution

1. 1

   Recognise the two-point form. The layout

   $\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$

   matches with $(x_1, y_1) = (362, -397)$ and $(x_2, y_2) = (366, -409)$: the numerator $y + 397$ is $y - (-397)$, and $-409 + 397$ is $y_2 - y_1$. So this is the line through those two points.

2. 2

   Simplify both denominators first.

   $-409 + 397 = -12, \qquad 366 - 362 = 4$

   so the equation is

   $\frac{y + 397}{-12} = \frac{x - 362}{4}$

   The slope is already visible: $m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-12}{4} = -3$.

3. 3

   Cross-multiply.

   $4(y + 397) = -12(x - 362)$

   Keep the negative sign attached to the $12$; dropping it flips the slope and is the most common error in this rearrangement.

4. 4

   Expand both sides.

   $4y + 1588 = -12x + 4344$

   Here $4 \times 397 = 1588$ and $12 \times 362 = 4344$.

5. 5

   Isolate y.

   $4y = -12x + 4344 - 1588 = -12x + 2756$

   $y = -3x + 689$

6. 6

   Check both original points lie on the line. At $x = 362$: $y = -1086 + 689 = -397 \ \checkmark$. At $x = 366$: $y = -1098 + 689 = -409 \ \checkmark$. The slope $-3$ agrees with the rise over run computed in step 2, so the conversion is correct.