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I'm doing homework and I've been given a line as the equation $7x-6y=5.$ I need to make a function $y = f(x)$ of this that corresponds to the equation.

What I know at this point is that I need to know what the $a$ and $b$ are in $y = ax + b.$ I also know of the formula to calculate $a$, which is $\,\dfrac{y_{b} - y_{a}}{ x_{b} - x_{a}}.\,$ I'm also wondering what a is called in English? It's "richtingscoëfficient" in Dutch.

I've tried to fill in the equation by using random numbers as x to calculate y, and using the formula $\dfrac{y_{b} - y_{a}}{ x_{b} - x_{a}}$ , but when I used the formula twice it turned out the answers were different from each other, which I don't understand, because it seemed logical to me that the answer had to be equal.

Please try to keep the answer simple, so do not use many symbols because I hardly know any yet.

Can someone correct my comment? I don't know what to fill in as tags, and how to put the formula's in the nice latex.

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If I understood it right, you need to express that equation in the form $y=f(x)$, is that right? –  Marra Jul 25 '13 at 0:43
    
yes thats correct –  user1534664 Jul 25 '13 at 0:44

3 Answers 3

up vote 2 down vote accepted

You can do it like this. Let's begin with the expression you have: $$ 7x-6y=5 $$ Adding the term $-7x$ to both sides, they remain equal and become $$ 7x-6y-7x=5-7x $$ that is $$ -6y=5-7x $$ Now let's multiply both sides of this equation by $-\dfrac{1}{6}$. We get: $$ y=\dfrac{7}{6}x-\dfrac{5}{6} $$ which is in the form you wanted. By the way, the number $\dfrac{7}{6}$ is called the slope of the line, which is a measure of its inclination.

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thanks dude, this is a brilliant answer –  user1534664 Jul 25 '13 at 1:02
    
Can I also divide it by -6, instead of -1/6? –  user1534664 Jul 25 '13 at 1:05
2  
I understand why you used fractions now, because else you end up with alot of decimal places :) –  user1534664 Jul 25 '13 at 1:07
    
:) thank you, I hope it was clear enough –  Marra Jul 25 '13 at 1:39

Try simple and avoid teacher jargon, it's used for make your remember it, but in fact they are not associated with something real.

Try to solve it as a equation where the unknown is y

$\begin{array}{l} 7x - 6y = 5\\ - 6y = 5 - 7x\\ 6y = 7x - 5\\ y = \frac{{7x - 5}}{6} = \frac{7}{6}x - \frac{5}{6} \end{array}$

Then:

$f(x) = \frac{7}{6}x - \frac{5}{6}$

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by teacher jargon u mean the formula's? –  user1534664 Jul 25 '13 at 0:47
    
No i mean the name. aThe formula is important, but you must understand what it mean. $\frac{{{y_a} - {y_b}}}{{{x_a} - {x_b}}}$ is in fact the rate of growth. For example i have 10\$ the 8th of month, and 20\$ the 18th, if i assume that i earn the value everyday, the money i earn by day: $\frac{{20\$ - 10\$ }}{{18 - 8}} = 1\$ $ –  John Math Jul 25 '13 at 0:48
    
thanks, I understand that formula, but the explanations in my assignments book are very short, so seeing the formula like that makes a lot of sense :) –  user1534664 Jul 25 '13 at 0:58

The $a$ is the $\textit{slope}$ of the linear function, which is just a measure of the rate of change of the line.

$7x-6y=5\implies-6y=-7x+5 \implies 6y=7x-5 \implies y=\dfrac{7x-5}{6}$

In this case, $\dfrac{7}{6}$ is the slope.

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