# What is the equation of the line passing through (96,72) and (19,4)?

May 7, 2018

The slope is 0.88311688312.

#### Explanation:

$\frac{{Y}_{2} - {Y}_{1}}{{X}_{2} - {X}_{1}}$ = $m$, the slope

(96, 72) $\left({X}_{1} , {Y}_{1}\right)$
(19, 4) $\left({X}_{2} , {Y}_{2}\right)$

$\frac{4 - 72}{19 - 96}$ = $m$

-68/-77 = $m$

Two negatives make a positive, so:

0.88311688312 = $m$

May 7, 2018

$y = = \frac{68}{77} x - \frac{984}{77}$

#### Explanation:

Recall;

$y = m x + c$

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

${y}_{2} = 4$

${y}_{1} = 72$

${x}_{2} = 19$

${x}_{1} = 96$

Inputing the values..

$m = \frac{4 - 72}{19 - 96}$

$m = \frac{- 68}{-} 77$

$m = \frac{68}{77}$

The new equation is;

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

Inputing their values..

$y - 72 = \frac{68}{77} \left(x - 96\right)$

$y - 72 = \frac{68 x - 6528}{77}$

Cross multiplying..

$77 \left(y - 72\right) = 68 x - 6528$

$77 y - 5544 = 68 x - 6528$

Collecting like terms..

$77 y = 68 x - 6528 + 5544$

$77 y = 68 x - 984$

Dividing through by $77$

$y = = \frac{68}{77} x - \frac{984}{77}$

May 7, 2018

Point-slope form: $y - 4 = \frac{68}{77} \left(x - 19\right)$

Slope-intercept form: $y = \frac{68}{77} x - \frac{984}{77}$

Standard form: $68 x - 77 y = 984$

#### Explanation:

First determine the slope using the slope formula and the two points.

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

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

I'm going to use $\left(19 , 4\right)$ as $\left({x}_{1} , {y}_{1}\right)$ and $\left(96 , 72\right)$ as $\left({x}_{2} , {y}_{2}\right)$.

$m = \frac{72 - 4}{96 - 19}$

$m = \frac{68}{77}$

Now use the slope and one of the points to write the equation in point-slope form:

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

where:

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

I'm going to use $\left(19 , 4\right)$ for the point.

$y - 4 = \frac{68}{77} \left(x - 19\right)$ $\leftarrow$ point-slope form

Solve the point-slope form for $y$ to get the slope-intercept form:

$y = m x + b$,

where:

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

$y - 4 = \frac{68}{77} \left(x - 19\right)$

Add $4$ to both sides of the equation.

$y = \frac{68}{77} \left(x - 19\right) + 4$

Expand.

$y = \frac{68}{77} x - \frac{1292}{77} + 4$

Multiply $4$ by $\frac{77}{77}$ to get an equivalent fraction with $77$ as the denominator.

$y = \frac{68}{77} x - \frac{1292}{77} + 4 \times \frac{77}{77}$

$y = \frac{68}{77} x - \frac{1292}{77} + \frac{308}{77}$

$y = \frac{68}{77} x - \frac{984}{77}$ $\leftarrow$ slope-intercept form

You can convert the slope-intercept form to the standard form:

$A x + B y = C$

$y = \frac{68}{77} x - \frac{984}{77}$

Multiply both sides by $77$.

$77 y = 68 x - 984$

Subtract $68 x$ from both sides.

$- 68 x + 77 y = - 984$

Multiply both sides by $- 1$. This will reverse the signs, but the equation represents the same line.

$68 x - 77 y = 984$ $\leftarrow$ standard form

graph{68x-77y=984 [-10, 10, -5, 5]}