Question

A line passes through the points (k +10, -2k-1) and (2,9) and has a y intercept of 10. Find the value of k and the equation of the line.

Answer 1

The y-intercept of 10 means a point (0,10) on the line.

We already have the point (2,9), so let's find the slope:

$\begin{gathered} m=\frac{\Delta y}{\Delta x} \\ m=-\frac{1}{2} \end{gathered}$

Now, we'll use the value of the slope to calculate k:

$\begin{gathered} m=\frac{\Delta y}{\Delta x} \\ m=\frac{-2k-1-9}{k+10-2} \\ \frac{-1}{2}=\frac{-2k-10}{k+8} \\ -k-8=-4k-20 \\ -k+4k=-20+8 \\ 3k=-12 \\ k=-4 \end{gathered}$

Now for the equation, we'll use the point (2,9)

$\begin{gathered} y-y1=m(x-x1) \\ y-9=-\frac{1}{2}(x-2) \\ y-9=-\frac{1}{2}x_{}+1 \\ y=-\frac{1}{2}x+10 \end{gathered}$

We could also consider we have the y-intercept, which is 10. So, b=10.

Using slope-intercept form, we'll have

y = -1/2m +10

$\begin{gathered} y-10=-\frac{1}{2}(x-0) \\ y-10=-\frac{1}{2}x \\ y=-\frac{1}{2}x+10 \end{gathered}$