$$ \left. \begin{array} { l } { ightarrow 2 \log ( x ) + \log ( y ) = 2 } \ { ightarrow \log ( x ) - \log ( y ) = 1 } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { ightarrow 2 \log ( x ) + \log ( y ) = 2 } \ { ightarrow \log ( x ) - \log ( y ) = 1 } \end{array} ight. $$

UpStudy AI · Accepted Answer

Para resolver el sistema de ecuaciones:

$$
\begin{cases}
2 \log(x) + \log(y) = 2 \
\log(x) - \log(y) = 1
\end{cases}
$$

**Paso 1:** A partir de la segunda ecuación:

$$
\log(x) - \log(y) = 1 \implies \log\left(\frac{x}{y}\right) = 1 \implies \frac{x}{y} = 10^1 = 10 \implies x = 10y
$$

**Paso 2:** Sustituimos $$ x = 10y $$ en la primera ecuación:

$$
2 \log(10y) + \log(y) = 2
$$

Descomponemos $$ \log(10y) $$:

$$
\log(10y) = \log(10) + \log(y) = 1 + \log(y)
$$

Por lo tanto:

$$
2(1 + \log(y)) + \log(y) = 2 \implies 2 + 2\log(y) + \log(y) = 2 \implies 3\log(y) = 0 \implies \log(y) = 0
$$

**Paso 3:** Resolvemos para $$ y $$:

$$
\log(y) = 0 \implies y = 10^0 = 1
$$

**Paso 4:** Encontramos $$ x $$ usando $$ x = 10y $$:

$$
x = 10 \times 1 = 10
$$

**Solución Final:**

$$
x = 10 \quad \text{y} \quad y = 1
$$
$$ x = 10 $$ y $$ y = 1 $$.

\( \left. \begin{array} { l } { \rightarrow 2 \log ( x ) + \log ( y ) = 2 } \\ { \rightarrow \log ( x ) - \log ( y ) = 1 } \end{array} \right. \)