Find the slope of the line that passes through the points (-1, -2) and (-4, -1).

Step-by-step solution

1. 1

   Label the points consistently. Take $(x_1,y_1)=(-1,-2)$ and $(x_2,y_2)=(-4,-1)$. The labelling is arbitrary — swapping the two points negates both numerator and denominator, so the slope is unchanged — but the same choice must be used in both parts of the fraction.

2. 2

   Compute the rise, minding the double negatives.

   $y_2-y_1=-1-(-2)=-1+2=1$

   Subtracting a negative adds; dropping that second sign gives $-3$ and a completely wrong answer.

3. 3

   Compute the run the same way.

   $x_2-x_1=-4-(-1)=-4+1=-3$

4. 4

   Form the quotient and place the sign.

   $m=\frac{1}{-3}=-\frac13$

   Conventionally the minus sign is written in front of the fraction rather than in the denominator, so the answer is $m=-\tfrac13$: the symbol is $-$, the numerator $1$, the denominator $3$.

5. 5

   Check by walking the line and by reversing the labels. Starting at $(-1,-2)$ and moving $3$ left ($\Delta x=-3$) should raise $y$ by $1$: $(-4,-1)$ ✓, which is the second point. Reversing the labels gives $\dfrac{-2-(-1)}{-1-(-4)}=\dfrac{-1}{3}=-\tfrac13$ — the same value ✓.