Full solution

2

. Find the value of

x

and

y

which satisfy the equations

\newline

\begin{array}{c} \frac{9^{x}}{3^{y}}=\frac{1}{9} \\ \log _{2} x=8^{\frac{1}{3}}-\log _{2} y \end{array}

Rewrite using exponents: Rewrite the first equation using properties of exponents. $\newline$ $(9^{x})/(3^{y}) = (1)/(9)$ $\newline$ Since $9 = 3^2$ , we can write $9^x$ as $(3^2)^x$ and $1/9$ as $3^{-2}$ . $\newline$ $(3^{2x})/(3^{y}) = 3^{-2}$
Combine left side: Use the property of exponents to combine the left side. $\newline$ $3^{2x - y} = 3^{-2}$ $\newline$ Since the bases are the same, the exponents must be equal. $\newline$ $2x - y = -2$
Rewrite using logarithms: Rewrite the second equation using properties of logarithms. $\newline$ $\log_{2}x = 8^{\frac{1}{3}} - \log_{2}y$ $\newline$ Since $8 = 2^3$ , we can write $8^{\frac{1}{3}}$ as $(2^3)^{\frac{1}{3}}$ . $\newline$ $\log_{2}x = 2 - \log_{2}y$
Combine right side: Use the property of logarithms to combine the right side. $\newline$ $\log_{2}x + \log_{2}y = 2$ $\newline$ Apply the product rule of logarithms: $\log_b(m) + \log_b(n) = \log_b(mn)$ . $\newline$ $\log_{2}(xy) = 2$
Convert to exponential: Convert the logarithmic equation to an exponential equation. $\newline$ $2^{\log_{2}(xy)} = 2^2$ $\newline$ $xy = 2^2$ $\newline$ $xy = 4$
Solve for x: Now we have a system of two equations: $\newline$ $2x - y = -2$ $\newline$ $xy = 4$ $\newline$ Let's solve for x using the second equation. $\newline$ $x = \frac{4}{y}$
Substitute $x$ into first: Substitute $x = \frac{4}{y}$ into the first equation. $2\left(\frac{4}{y}\right) - y = -2$ $\frac{8}{y} - y = -2$ Multiply through by $y$ to clear the fraction. $8 - y^2 = -2y$
Rearrange to quadratic: Rearrange the equation to form a quadratic equation. $\newline$ $y^2 - 2y - 8 = 0$
Factor the equation: Factor the quadratic equation. $\newline$ $(y - 4)(y + 2) = 0$
Solve for $y$ : Set each factor equal to zero and solve for $y$ . $y - 4 = 0$ or $y + 2 = 0$ $y = 4$ or $y = -2$
Substitute $y = 4$ : Substitute $y = 4$ into $xy = 4$ to find $x$ . $x(4) = 4$ $x = 1$
Substitute $y = -2$ : Substitute $y = -2$ into $xy = 4$ to find $x$ . $x(-2) = 4$ $x = -2$