Originally Posted by

**Soroban** Hello, bonapartist!

Here's some help . . .

First, get rid of the fractions.

. . $\displaystyle \begin{array}{ccccc}\text{Multiply [1] by 8:} & x + 6y & = & 38 & [3] \\ \text{Multiply [2] by 4:} & \text{-}2x + 3y & = & 2 & [4]\end{array}$

Solve [3] for $\displaystyle x\!:\;\;x \:=\:38-6y$

Substitute into [4]: .$\displaystyle \text{-}2(38-6y) + 3y \:=\:2\quad\Rightarrow\quad\text{-}76 + 12y + 3y \:=\:2$

. . $\displaystyle 15y \,=\,78\quad\Rightarrow\quad\boxed{ y \,=\,\frac{26}{5}}$

Substitute into [3]: .$\displaystyle x + 6\left(\frac{26}{5}\right) \:=\:38$

. . $\displaystyle x \:=\:38 - \frac{156}{5}\quad\Rightarrow\quad\boxed{ x\,=\,\frac{34}{5}}$

What happened is not your fault . . .

$\displaystyle \begin{array}{cccc}\text{Divide [2] by -3:} & \text{-}x - y & = & 2 \\ \text{Add [1]:} & x + y & = & 4 \\

\text{and we get:} & 0 & = & 6 \end{array}$

We get a statement which is obviously not true.

This means that the sytem has **no** solution . . . it is *inconsistent.*