# Question 32f04

Mar 24, 2017

Set $b$ to any value you choose then

set $\text{ "a=-4b" }$ and set$\text{ } c = - 7 b$

#### Explanation:

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Given:$\text{ } a x + b y = c$

Subtract $\textcolor{red}{a x}$ from both sides

color(green)(ax color(red)(-ax) + by" " =" "c color(red)(-ax)

$\textcolor{g r e e n}{0 + b y \text{ "=" } c - a x}$

Divide both sides by $\textcolor{red}{b}$

color(green)((by)/(color(red)(b))" "=" "c/(color(red)(b))-(ax)/(color(red)(b))#

But $\frac{b}{b} = 1$

$\textcolor{g r e e n}{y \text{ "=" } \frac{c}{b} - \frac{a x}{b}}$

Changing the order to match convention we have:

$\textcolor{g r e e n}{y = - \frac{a}{b} x + \frac{c}{b}}$

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Compare $y = - \frac{a}{b} x + \frac{c}{b} \text{ }$ to $\text{ } y = 4 x - 7$

So we have:

$- \frac{a}{b} = 4 \text{ " ->" " a/b=-4" "->" } a = - 4 b$

$\frac{c}{b} = - 7 \text{ "->" } c = - 7 b$

This gives us a sort of 'open book' about choices for $a \mathmr{and} c$

Example: Let $b = 3$ or any value you so choose, then we have:

$a = - 4 \left(3\right) = - 12$
$c = - 7 \left(3\right) = - 21$

Thus $a x + b y = c \text{ implies that } - 12 x + 3 y = - 21$

$3 y = + 12 x - 21$

$y = \frac{12}{3} x - \frac{21}{3} \to 4 x - 7 \textcolor{red}{\text{ IT WORKS!}}$