<s>[INST] Help to solve and understand Simultaneous equations. Could anyone help me solve this simultaneous equation:
$$6x + 7y = 12.5\tag1$$
$$7x + 5y = 14\tag2$$
I have watched loads of videos on YouTube, and still don't understand how to do it. I am trying to learn it. And practice with different equations. But don't understand how to do it. So any help will be much appreciated. 
 [/INST] We have \begin{align}6x + 7y &= 12.5\tag{1}\\
7x + 5y &= 14\tag{2}\end{align}
One way to solve this, would be to rearrange equation $(1)$ for $x$:
\begin{align}6x + 7y &= 12.5\\
6x&=12.5-7y\\
x&=\frac{12.5-7y}{6}\end{align}
We can then put this into equation $(2)$ and solve for $y$:
\begin{align}7x + 5y &= 14\\
7\left(\frac{12.5-7y}{6}\right)+5y&=14\\
\frac{87.5-49y}{6}+5y&=14\\
87.5-49y+30y&=84\\
87.5-19y&=84\\
19y&=3.5\\
y&=\frac{3.5}{19}\\
&=\frac{7}{38}\end{align}
Finally we can put this into equation $(1)$ to find that 
\begin{align}6x + 7\left(\frac{7}{38}\right) &= 12.5\\
6x+\frac{49}{38}&=12.5\\
6x&=12.5-\frac{49}{38}\\
6x&=\frac{213}{19}\\
x&=\frac{71}{38}\end{align}
We can then check this using equation $(2)$:
\begin{align}7x + 5y &=7\left(\frac{71}{38}\right)+5\left(\frac{7}{38}\right)\\
&=\frac{497}{38}+\frac{35}{38}\\
&=\frac{532}{38}\\
&=14\end{align}
Therefore we can be confident that the answer is $x=\dfrac{71}{38}$ and $y=\dfrac{7}{38}$
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