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=(\Delta y)/(\Delta x) \\ m=-(1)/(2) \end{gathered}`

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

`\begin{gathered} m=(\Delta y)/(\Delta x) \\ m=(-2k-1-9)/(k+10-2) \\ (-1)/(2)=(-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=-(1)/(2)(x-2) \\ y-9=-(1)/(2)x_{}+1 \\ y=-(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=-(1)/(2)(x-0) \\ y-10=-(1)/(2)x \\ y=-(1)/(2)x+10 \end{gathered}`