Question

The line m is parallel to the line y = 1/2x + 7 and passes through the point (2k, 3k - 3) and (k - 1, 2k - 2). What is the value for k?

Answer 1

Answer:

$k = 3$.

Step-by-step explanation:

The equation of the line $y = (1/2) \, x + 7$ is in the slope-intercept form: $y = m\, x + b$, where $m$ denotes the slope of this line.

Hence, the slope of the line $y = (1/2) \, x + 7$ is $(1/2)$.

In a plane, two lines are parallel if and only if their slope is the same. Hence, the slope of any line parallel to this line would also be $(1/2)$.

The slope of a non-vertical line that goes through two distinct points $(x_{0},\, y_{0})$ and $(x_{1},\, y_{1})$, where $x_{0} \ne x_{1}$, is:

$\displaystyle m = \frac{y_{1} - y_{0}}{x_{1} - x_{0}}$.

In this question, the two points are $(2\, k,\, 3\, k - 3)$ and $(k - 1,\, 2\, k - 2)$. Assume that $2\, k \ne (k -1)$. The slope of the line that goes through these two points would be:

$\begin{aligned}& \frac{y_{1} - y_{0}}{x_{1} - x_{0}} \\ =\; & \frac{(2\, k - 2) - (3\, k - 3)}{(k - 1) - 2\, k} \\ =\; & \frac{2\, k - 2 - 3\, k + 3}{k - 1 - 2\, k} \\ =\; & \frac{-k + 1}{-k - 1} \\ =\; & \frac{k - 1}{k + 1}\end{aligned}$.

Since this line is parallel to $y = (1/2) \, x + 7$, the slope of this line would also be $(1/2)$. Therefore:

$\displaystyle \frac{k - 1}{k + 1} = \frac{1}{2}$.

$2\, (k - 1) = k + 1$.

$k = 3$.

Verify that $2\, k \ne (k - 1)$.