Simultaneous equation other possibilities but how? ($14x - 5y + 14 = 179, \ -14x + 5y + 5 = -160$) Can you please help me identify where I went wrong, for the following question, when I plug in my answers using substitution method I get the correct answer, but I found out that it could also be $15$ and $9$ but how do you get that.
$$14x - 5y + 14 = 179  \\
-14x + 5y + 5 = -160.$$
I rearranged the first equation,
$14x - 5y + 14 = 179\\ 
14x = 5y -14 +179\\
x = \frac{5}{14}y + \frac{165}{14}.$
Then using the second equation, I solve for $y$.
$-14x+5y+5=-160\\
5y+5=160+14x\\
5y=-165+14x\\
y=-33+\frac{14}{5}x$
Sub the $x$ value which we got before
$y=-33+14/5(165/14+5y/14) \Rightarrow\\
y=1.$
As $y=1$, sub this into the second equation,
$-14x + 5(1)+5 =-160\\
-14x= -170\\
x= 12.142857$
to 5 decimal places $= \ 12.14286$
$14(12.14286)-5(1)+14 = 179.00004\\
-14(12.14286)+5(1)+5 = -160.00004$
Where did I go wrong?
 A: First off, you made a mistake when finding the value of $y$. $y=-33+\frac{14}{5}(\frac{165}{14}+\frac{5}{14}y)$ is correct, but if you expand everything you get $y = -33 + 33 +y$. This reduced to $0=0$ which, while correct, is certainly not very useful.
The problem here is that you have these two equations:
$$14x - 5y + 14 = 179 \\  -14x + 5y + 5 = -160
$$
Rearrange to separate unkowns and numbers:
$$14x-5y = 165 \\
-14x+57 = -165$$
Multiply the bottom equation by $-1$:
$$
14x-5y = 165 \\
14x-5y = 165 
$$
See? What looked like two equations is actually only one written differently. From this equation you could, for example, solve for $y$:
$$y= \frac{14}{5}x-33$$
This tells you that you have infinite solutions; just pick a value of $x$ and you'll get a suitable value of $y$ that fulfils the equation. If you take $x=15$, you get $y = \frac{14}{5}\times 15 - 33 = 42-33 = 9$.
A: In the comments, OP asks, "Can any one show how you can get 15 and 9 as the answers?" 
First, as Javier points out, you show that your two equations are the same, so you really just have to find a solution to a single equation, $$14x-5y=165.$$ Second, you make an executive decision that you are only interested in whole number answers (or maybe the person who gave you the problem tells you that she is only interested in whole number answers). In that case, since $165$ is a multiple of $5$, and $5y$ is a multiple of $5$, $14x$ must also be a multiple of $5$; and since $14$ is not a multiple of $5$, which is a prime number, $x$ must be a multiple of $5$. That is, $x=5Q$ for some whole number $Q$. So our equation becomes, $$(14)(5Q)-5y=165,$$ and we can divide through by $5$ to get $$14Q-y=33,$$ which we can rewrite as $$y=14Q-33.$$ Now you can let $Q$ be any number you like, and this formula will tell you what $y$ has to be (and $x=5Q$ will tell you what $x$ has to be). Since you decided that you want whole numbers, you had better pick $Q$ to be a whole number, and you had better pick it large enough to make $14Q-33$ a whole number. A little experimentation (or dividing $33$ by $14$) will convince you that the smallest number that works for $Q$ is $Q=3$. Then $x=5Q=15$, and $y=14Q-33=9$, and that's how you can get 15 and 9 as answers. 
