Full solution

\begin{array}{l}10^{x+2 y}=5 \\ 10^{3 x+4 y}=50\end{array}

Convert to Logarithmic Form: Step $1$ : Convert the equations to logarithmic form to simplify the exponents. $\newline$ - Equation $1$ : $10^{x+2y} = 5$ becomes $\log_{10}(5) = x + 2y$ $\newline$ - Equation $2$ : $10^{3x+4y} = 50$ becomes $\log_{10}(50) = 3x + 4y$
Calculate Logarithmic Values: Step $2$ : Calculate the logarithmic values. $\newline$ - $\log_{10}(5) \approx 0.6990$ $\newline$ - $\log_{10}(50) = \log_{10}(5\times10) = \log_{10}(5) + \log_{10}(10) = 0.6990 + 1 = 1.6990$
Substitute Values into Equations: Step $3$ : Substitute the logarithmic values back into the equations. $\newline$ - $0.6990 = x + 2y$ $\newline$ - $1.6990 = 3x + 4y$
Solve System of Equations: Step $4$ : Solve the system of equations using elimination or substitution. $\newline$ - Multiply the first equation by $3$ : $3 \times 0.6990 = 3x + 6y$ $\newline$ - Subtract the first new equation from the second original equation: $\newline$ $1.6990 - 3 \times 0.6990 = 3x + 4y - (3x + 6y)$ $\newline$ $1.6990 - 2.0970 = -2y$ $\newline$ $0.6020 = -2y$ $\newline$ $y = -0.3010$
Find Values of x and y: Step $5$ : Substitute the value of y back into one of the original equations to find x. $\newline$ - $0.6990 = x + 2(-0.3010)$ $\newline$ - $0.6990 = x - 0.6020$ $\newline$ - $x = 0.6990 + 0.6020$ $\newline$ - $x = 1.3010$