# Parallel lines passing through $(d, 3)$ & $(-2,1)$ and $(5,d)$ & $(1,0)$.

There are two parallel lines with one passing through $(d, 3)$ & $(-2,1)$ and the second line passes through $(5,d)$ & $(1,0)$.

Find the two values for $d$.

I found one value would be $-4$. But how would I do this to find the second value?

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If two lines are parallel, then the slope of both the lines are equal.

If a line passes through the points, $(x_1,y_1)$ and $(x_2,y_2)$, then the slop of the line is given by $$\text{Slope } = \dfrac{y_1 - y_2}{x_1 - x_2}$$

The first line passes through the points $(d,3)$ and $(-2,1)$. Hence, the slope of the first line is $$\dfrac{3-1}{d-(-2)}$$

The second line passes through the points $(5,d)$ and $(1,0)$. Hence, the slope of the second line is $$\dfrac{d-0}{5-1}$$

Since the lines are parallel, the slope are equal. Equate the two to get $$\dfrac2{d+2} = \dfrac{d}4 \implies d^2+2d = 8 \implies d^2 +2d-8 = 0$$ Can you now finish it off?

$$d^2 + 2d - 8 = 0 \implies (d+4)(d-2) = 0 \implies d= 2 \text{ or } -4$$
@Michael Yes. $2$ and $-4$ are the desired slopes. – user17762 Jul 1 '12 at 0:18