Question

Find the value of k so that the three points lie on the same line. Write the equation of the line in point-slope form.
(1,-2)
(-2,4)
(4,k)

Answer 1

k= -8
y+8 = -2 (x - 4)

Answer 2

How did you come up with k=-8?

Answer 3

Can someone explain how to solve this?

Answer 4

Since all three points are on the same line, the slope of the segment connecting any two points will be the same.

So, you know that
\[\large\underbrace{\frac{4-(-2)}{-2-1}}_{\text{slope between first and second point}}=\underbrace{\frac{k-4}{4-(-2)}}_{\text{slope between second and third point}}\]
Solve the equation for \(k\):
\[\begin{align*}\frac{4-(-2)}{-2-1}&=\frac{k-4}{4-(-2)}\\
\frac{6}{-3}&=\frac{k-4}{6}\\
-2&=\frac{k-4}{6}\\
-12&=k-4\\
-12+4&=k\\
-8&=k
\end{align*}\]

Answer 5

@JoStudy, to find the equation of the line, you use the point-slope formula:
\[y-y_0=m(x-x_0)\]
where \(m\) is the slope and \((x_0,y_0)\) is any point that you know is on the line.

We've already found out that \(m=-2\) (third line in the work above). Now pick any of the three points for \((x_0,y_0)\), then plug it all into the equation.

Answer 6

Thanks!