What are the intercepts of 8x-5y=2?

Nov 20, 2015

$\textcolor{g r e e n}{y = \frac{8}{5} x - \frac{2}{5}}$
The explanation shows the underlying principles behind the short cuts people show you!!

Explanation:

$\textcolor{b l u e}{\underline{S t e p \textcolor{w h i t e}{x} 1}}$

Add $\textcolor{b l u e}{5 y}$ to both sides

color(brown)((8x-5y) color(blue)(+5y) =( 2)color(blue)(+5y)

I am using the brackets to show what is being altered or grouping to make understanding easier. They serve no other purpose!

$8 x + \left(\textcolor{b l u e}{5 y} - 5 y\right) = 2 + \textcolor{b l u e}{5 y}$

$8 x + 0 = 2 + 5 y \textcolor{g r e e n}{\text{ This action has made the y-term positive}}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)(underline(Stepcolor(white)(x)2)

Subtract 2 from both sides

$\textcolor{b r o w n}{\left(8 x\right) \textcolor{b l u e}{- 2} = \left(5 y + 2\right) \textcolor{b l u e}{- 2}}$

$8 x - 2 = 5 y + \left(2 - 2\right)$

$8 x - 2 = 5 y + 0 \textcolor{w h i t e}{\times} \textcolor{g r e e n}{\text{This action moved the 2 to the other side of =}}$

Due to convention rewrite with the target variable on the left:

$5 y = 8 x - 2$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)(underline(Stepcolor(white)(x)3)

Divide both sides by $\textcolor{b l u e}{5}$. Note that this is the same as $\times \frac{1}{5}$

$\frac{\textcolor{b r o w n}{5 y}}{\textcolor{b l u e}{5}} = \frac{\textcolor{b r o w n}{8 x - 2}}{\textcolor{b l u e}{5}}$

$\frac{5}{5} \times y = \frac{8}{5} x - \frac{2}{5}$

But $\frac{5}{5} = 1$ giving

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)(underline(Solution)

$\textcolor{g r e e n}{y = \frac{8}{5} x - \frac{2}{5}}$