How do you find the equation of the line that goes through (3, -5) and (5, 4)?

Sep 1, 2017

See below

Explanation:

The method is called Two point form ( At least in my school )

The method is

let
3 = ${x}_{1}$
-5 = ${y}_{1}$
5 = ${x}_{2}$
4 = ${y}_{2}$

To find an equation passing through these two points,

$\left(y - {y}_{1}\right) = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} \cdot \left(x - {x}_{1}\right)$

Where y and x are all the points on the equation, that has to be found out.

Now plugging in the values,

$\left(y + 5\right) = \frac{4 + 5}{5 - 3} \cdot \left(x - 3\right)$

$\left(y + 5\right) = \frac{9}{2} \cdot \left(x - 3\right)$

$\left(y + 5\right) 2 = 9 \left(x + 3\right)$

$2 y + 10 = 9 x + 18$

$9 x - 2 y + 18 - 10 = 0$

$9 x - 2 y + 8 = 0$

Sep 1, 2017

$y = \frac{9}{2} x - \frac{37}{2}$

Explanation:

Begin by finding the slope via the slope formula: $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

If we let,
$\left(3 , - 5\right) \to \left(\textcolor{red}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right) \mathmr{and} \left(5 , 4\right) \to \left(\textcolor{red}{{x}_{2}} , \textcolor{b l u e}{{y}_{2}}\right)$

Then,

$m = \frac{\textcolor{b l u e}{4 - \left(- 5\right)}}{\textcolor{red}{5 - 3}} = \frac{9}{2}$

Now that we have the slope, we can find the equation of the line by using the point-slope formula:

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

Where $m$ is the slope and $\left({x}_{1} , {y}_{1}\right)$ is a point on the function. We can use any of the two coordinates given. I will use $\left(5 , 4\right)$ as my $\left({x}_{1} , {y}_{1}\right)$

Thus, the equation of the line is...

$y - 4 = \frac{9}{2} \left(x - 5\right) \leftarrow$ Equation in point-slope form

We can rewrite the equation above in $y = m x + b$ if desired by solving t=for the variable $y$

$y - 4 = \frac{9}{2} x - \frac{45}{2}$

$y \cancel{- 4 + 4} = \frac{9}{2} x - \frac{45}{2} + 4$

$y = \frac{9}{2} x - \frac{45}{2} + 4 \left(\frac{2}{2}\right)$

$y = \frac{9}{2} x - \frac{45}{2} + \frac{8}{2}$

$y = \frac{9}{2} x - \frac{37}{2} \leftarrow$ Equation in $y = m x + b$ form