Question

can you help me solve thus?

Answer 1

First, we check that the three points are collinear.

Given three points (x1,y1), (x2,y2), and (x3,y3), then the three points are collinear if the following condition is satisfied.

$\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_1}{x_3-x_1}$

In our case, we can set

(x1,y1) =(-2,0), (x2,y2) = ( 4, 18)and (x3,y3) = (5, 21)

$\begin{gathered} \text{ Then, we have} \\ \frac{y_2-y_1}{x_2-x_1}=\frac{18-0}{4-(-2)}=\frac{18}{4+2}=\frac{18}{6}=3 \end{gathered}$

And

$\frac{y_3-y_1}{x_3-x_1}=\frac{21-0}{5-(-2)}=\frac{21}{5+2}=\frac{21}{7}=3$

Hence

$\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_1}{x_3-x_1}$

Therefore, the points (-2,0), ( 4, 18), and (5, 21) are collinear.

Given two points (x1,y1) and (x2,y2) on the Cartesian plane, then the gradient, m, of the line that passes through them is given by

$m=\frac{y_2-y_1}{x_2-x_1}$

Then, the gradient of the line passing through (-2,0), ( 4, 18) is 3

Given a point (x1,y1) on the Cartesian plane, and a line that passes through the point with gradient, m, then the equation of the line is given by

$y-y_1=m(x-x_1)$

Therefore,

$y-0=3(x-(-2)=3(x+2)$

That is the equation is

$y=3x+6$