Find the equation of the line through the points (-6, 1) and (-3, 7).

Step-by-step solution

1. 1

   Compute the slope carefully with negative $x$-values. With $(x_{1},y_{1})=(-6,1)$ and $(x_{2},y_{2})=(-3,7)$:

   $m=\frac{7-1}{-3-(-6)}=\frac{6}{-3+6}=\frac{6}{3}=2$

   The run is $-3-(-6)=3$, not $-9$: subtracting a negative adds. Getting $-9$ here would give the wrong slope $-\tfrac23$.

2. 2

   Use point-slope form. With the point $(-6,1)$:

   $y-1=2\left(x-(-6)\right)=2(x+6)$

3. 3

   Distribute and solve for $y$.

   $y-1=2x+12\ \Longrightarrow\ y=2x+13$

4. 4

   Interpret the intercept. The $y$-intercept $13$ is far above both given $y$-values ($1$ and $7$) because the line is steep and both points sit well to the left of the $y$-axis: travelling from $x=-3$ to $x=0$ climbs another $3\times 2=6$ units, from $7$ up to $13$.

5. 5

   Verify with both points. At $x=-6$: $2(-6)+13=1$ ✓. At $x=-3$: $2(-3)+13=7$ ✓. Both check out, so $y=2x+13$ is the line.