Answer
(a) Process described below, in the "work step by step" box.
(b)
$x = -2 + 5t$
$y = 7 - 8t$
Work Step by Step
(a)
1. The equation that describes a general line that pass through $(x_1,y_1)$ and $(x_2,y_2)$ is:
$(y - y_1) = m(x - x_1)$
$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
$y - y_1 = (y_2-y_1) \frac{x - x_1}{x_2 - x_1}$
2. Considering only the line segment that joins these points, $x$ will go from $x_1$ to $x_2$.
$\frac{x_1-x_1}{x_2-x_1} = 0$
$\frac{x_2 - x_1}{x_2 - x_1} = 1$
Therefore, we can substitute $t$ for the fraction: $\frac{x-x_1}{x_2-x_1}$. And we will have: $(0 \leq t \leq 1)$
$y - y_1 = (y_2 - y_1)t$
$y = y_1 + (y_2 - y_1) t$
3. Now, solve for $x$:
$(y - y_1) \frac{x_2 - x_1}{y_2 - y_1} = x - x_1$
- Rearranging:
$\frac{y - y_1}{y_2 - y_1}(x_2 -x_1) = x - x_1$
- Repeat the same we did on step 2:
$(x_2-x_1)t = x- x_1$
$(x_2 - x_1)t + x_1 = x$
$x = x_1 (x_2 - x_1)t$
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(b)
$x = (-2) + (3 - (-2))t = -2 + 5t$
$y = 7 + ((-1) - 7) t = 7 - 8t$
