# What is the slope of the line passing through the following points:  (1/3, 2/5 ) , (-3/4, 5/3)?

May 8, 2017

Gradient (slope) $\to - \frac{76}{65}$

Negative means it slopes down reading left to right.

#### Explanation:

Have a look at https://socratic.org/s/aEw6Hquc

It uses different values but it has quite an extensive explanation.

Set point 1 as $_ {P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(- \frac{3}{4} , \frac{5}{3}\right)$
Set point 2 as ${P}_{2} \to \left({x}_{2} , {y}_{2}\right) = \left(\frac{1}{3} , \frac{2}{5}\right)$

When determining the gradient you read left to right on the x-axis

So as ${x}_{1} = - \frac{3}{4}$ it comes before ${x}_{2} = + \frac{1}{3}$

So the change in $x$ reading left to right is ${x}_{2} - {x}_{1}$

Also the change in $y$ reading left to right on the x-axis is$\textcolor{w h i t e}{.} {y}_{2} - {y}_{1}$

Thus the gradient is:

$\left(\text{change in y")/("change in x}\right) \to \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{\frac{2}{5} - \frac{5}{3}}{\frac{1}{3} - \left(- \frac{3}{4}\right)} = \frac{\frac{2}{5} - \frac{5}{3}}{\frac{1}{3} + \frac{3}{4}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Consider just the top (numerator) } \to \frac{2}{5} - \frac{5}{3}}$

color(green)([2/5color(red)(xx1)]-[5/3color(red)(xx1)]" "=" "[2/5color(red)(xx3/3)]-[5/3color(red)(xx5/5)]

$\text{ "color(green)(" } \left[\frac{6}{15}\right] - \left[\frac{25}{15}\right]$

$\text{ } \textcolor{g r e e n}{- \frac{19}{15}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Consider just the bottom (denominator) } \to \frac{1}{3} + \frac{3}{4}}$

color(green)([1/3color(red)(xx1)]+[3/4color(red)(xx1)]" "=" "[1/3color(red)(xx4/4)]+[3/4color(red)(xx3/3]]

" "color(green)([4/12]+[9/12]

$\text{ } \textcolor{g r e e n}{\frac{13}{12}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Putting it all together}}$

("change in y")/("change in x")" "=" "(color(white)(.)-19/15color(white)(.))/(13/12)

This is the same as: $\text{ } - \frac{19}{15} \times \frac{12}{13} = - 1 \frac{11}{65} \to - \frac{76}{65}$

Checking with a graph: 