Question

Find the equation of the line in standard form that passes through the following points. Eliminate anyfractions and simplify your answer.(4, -8) and (9, 11)

Answer 1

We want to find the equation of the line that passes through the points:

(4 , -8) and (9 , 11)

First, we're going to find the slope between these points using the fact that:

If we have two points that lie on a line:

`(x_1,y_1)\text{ and }(x_2,y_2)`

The slope between them can be found using the formula:

`m=(y_2-y_1)/(x_2-x_1)`

If we replace our values:

`\begin{gathered} (x_1,y_1)=(4,-8) \\ (x_2,y_2)=(9,11) \\ x_1=4 \\ x_2=9 \\ y_1=-8 \\ y_2=11 \end{gathered}`

The slope will be:

`m=(11-(-8))/(9-4)=(11+8)/(5)=(19)/(5)`

Now, we could apply the point-slope equation. This equation tells us that we can find the equation of the line if we got a point (x1,y1) on the line, and the slope m:

`y=y_1+m(x-x_1)`

Replacing our values:

`\begin{gathered} y=-8+(19)/(5)(x-4) \\ y=-8+(19)/(5)x-(76)/(5) \\ y=(19)/(5)x-(116)/(5) \end{gathered}`

This, is the general form. We want to express the last equation as a standard form like this:

`Ax+By=C`

If we re-write:

`\begin{gathered} y=(19x-116)/(5) \\ \\ 5y=19x-116 \\ 19x-5y=116 \end{gathered}`

Therefore, the standard for the equation of the line that passes through (4 , -8) and (9, 11) is:

19x-5y=116