Easier way to solve a LES (only pen/paper, no calculator)

Question

I want to solve the following Linear Equation System with only pen and paper;

$$ 470 = x_A - \frac{3}{10}x_B \tag{1} $$ $$ 940 = x_B - \frac{2}{10}x_A \tag{2} $$

I attempt to solve for $x_A$ by inserting equation (2) into (1) and rewriting it, etc. I got something like;

$$ x_A=470 +\frac{3}{10}(940+\frac{2}{10}x_A) \Rightarrow \dots \Rightarrow x_A\frac{94}{100}=470+\frac{3}{10}940=470+\frac{2820}{10} $$

Next I try to isolate $x_A$ and simplify the expression;

$$ \dots \Rightarrow x_A =\frac{100}{94}(470+\frac{2820}{10})=\frac{100}{94}(470+282)=\frac{100}{94}752 $$

This is where I get stuck. Have I dug myself into a difficult hole when there is a better/more efficient approach to solving this LES? Or am I just missing a good method to solve $x_A=\frac{100}{94}\cdot752$ to get it to 800?

Any insight into "how to think" here and how you solve it with just pen and paper is much appreciated.

(Solution: $x_A=800$, $x_B=1100$)

Answer 1

First notice $94=47\cdot 2$. You can simplify before adding/multiplying: $$x_A\frac{94}{100}=470+\frac{3}{10}940\color{red}{=470+\frac{2820}{10}} \Rightarrow \\ x_A\frac{47\cdot 2}{100}=47\cdot 10+\frac3{10}\cdot 47\cdot 20 \Rightarrow \\ x_A\frac2{100}=10+\frac{60}{10} \Rightarrow \\ 2x_A=1000+600 \Rightarrow \\ x_A=800.$$

Answer 2

What are your problems ?

$ x_A=\frac{100}{94}\cdot752$ is correct, since $752:94=8.$

Answer 3

May be, it could be simpler to do $$470 = x_A - \frac{3}{10}x_B \implies4700=10x_A-3x_B\tag{1}$$

$$940 = x_B - \frac{2}{10}x_A \implies9400=10x_B-2x_A\tag{2}$$ Multiply $(1)$ by $10$ and $(2)$ by $3$ and add them together; this gives immediately $$75200=94 x_A\implies x_A=\frac{75200}{94}=800$$ Plug this in $(2)$ to get $x_B$