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

##### 1 Answer
Mar 16, 2018

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:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} ,$

where:

$m$ is the slope, $\left({x}_{1} , {y}_{1}\right)$ is one point, and $\left({x}_{2} , {y}_{2}\right)$ is the second point.

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

$m = \frac{- 10 - 0}{25 - 0}$

Simplify.

$m = - \frac{10}{25}$

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

$m = - \frac{10 \div 5}{25 \div 5}$

$m = - \frac{2}{5}$

The slope is $- \frac{2}{5}$.

Point-slope form

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

$y - {y}_{1} = m \left(x - {x}_{1}\right) ,$

where:

$m$ is the slope, and $\left({x}_{1} , {y}_{1}\right)$ is the point. You can use either point from the given information. I'm going to use $\left(0 , 0\right)$. Again, you can use the other point. It will end up the same, but take more steps.

$y - 0 = - \frac{2}{5} \left(x - 0\right)$ $\leftarrow$ point-slope form

Slope-intercept form

Now we can determine the slope-intercept form:

$y = m x + b ,$

where:

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

Solve the point-slope form for $y$.

$y - 0 = - \frac{2}{5} \left(x - 0\right)$

$y = - \frac{2}{5} x$ $\leftarrow$ slope-intercept form $\left(b = 0\right)$

Standard form

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

$A x + B y = C ,$

where:

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

$y = - \frac{2}{5} x$

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

$5 y = \frac{- 2 x}{\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}}} ^ 1 {\left(\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}}\right)}^{1}$

$5 y = - 2 x$

Add $2 x$ to both sides.

$2 x + 5 y = 0$ $\leftarrow$ standard form

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