Question

What is the equation of the line between `(0,0)` and `(25,-10)`?

Answer 1

This answer will show you how to determine the slope of a line, and how to determine the point-slope, slope-intercept, and standard forms of a linear equation.

Explanation:

Slope

First determine the slope using the formula:

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

where:

`m` is the slope, `(x_1,y_1)` is one point, and `(x_2,y_2)` is the second point.

Plug in the known data. I am going to use `(0,0)` as the first point, and `(25,-10)` as the second point. You can do the opposite; the slope will be the same either way.

`m=(-10-0)/(25-0)`

Simplify.

`m=-10/25`

Reduce by dividing the numerator and denominator by `5`.

`m=-(10-:5)/(25-:5)`

`m=-2/5`

The slope is `-2/5`.

Point-slope form

The formula for the point-slope form of a line is:

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

where:

`m` is the slope, and `(x_1,y_1)` is the point. You can use either point from the given information. I'm going to use `(0,0)`. Again, you can use the other point. It will end up the same, but take more steps.

`y-0=-2/5(x-0)` `larr` point-slope form

Slope-intercept form

Now we can determine the slope-intercept form:

`y=mx+b,`

where:

`m` is the slope, and `b` is the y-intercept.

Solve the point-slope form for `y`.

`y-0=-2/5(x-0)`

`y=-2/5x` `larr` slope-intercept form `(b=0)`

Standard form

We can convert the slope-intercept form into the standard form for a linear equation:

`Ax+By=C,`

where:

`A` and `B` are integers, and `C` is the constant (y-intercept)#

`y=-2/5x`

Eliminate the fraction by multiplying both sides by `5`.

`5y=(-2x)/color(red)cancel(color(black)(5))^1(color(red)cancel(color(black)(5)))^1`

`5y=-2x`

Add `2x` to both sides.

`2x+5y=0` `larr` standard form

graph{y=-2/5x [-10, 10, -5, 5]}