[Systems of Linear Equations]

Solve the system of equations with $n$ unknowns $\frac{x_{1}}{a_{1}}=\frac{x_{2}}{a_{2}}=\ldots=\frac{x_{n}}{a_{n}}, x_{1}+x_{2}+\ldots+x_{n}=A$.
Let each of the equal ratios be denoted by the letter $t$. Then $x_{1}=a_{1} t, x_{2}=a_{2} t, \ldots, x_{n}=a_{n} t$. Substituting these expressions into the last equation of the system, we get $t\left(a_{1}+a_{2}+\ldots+a_{n}\right)=A$, that is, $t=\frac{A}{a_{1}+\ldots+a_{n}}$.

## Answer

When $a_{1}+\ldots+a_{n} \neq 0 \quad x_{1}=\frac{A a_{1}}{a_{1}+\ldots+a_{n}}, \ldots, x_{n}=\frac{A a_{n}}{a_{1}+\ldots+a_{n}} ;$ when $a_{1}+\ldots+a_{n}=A=0\left(x_{1}, \ldots, x_{n}\right)$ $=\left(a_{1} t, a_{2} t, \ldots, a_{n} t\right)$, where $t-$ is any number; when $a_{1}+\ldots+a_{n}=0, A \neq 0$ there are no solutions.