If x and y are real numbers such that 2^x \cdot 3^y = 24 and 3^x \cdot 2^y = 54, what is t… — ACT

If $x$ and $y$ are real numbers such that $2^x \cdot 3^y = 24$ and $3^x \cdot 2^y = 54$ , what is the value of $x^2 + y^2$ ?

Answer: 10

Answer

Multiplying the two equations gives

(2^x \cdot 3^y)(3^x \cdot 2^y) = 24 \cdot 54 \implies 6^{x+y} = 1296 = 6^4

, which yields

x + y = 4

. Dividing the first equation by the second equation gives \frac{2^x \cdot 3^y}{3^x \cdot 2^y} = \frac{24}{54} \implies (\frac{2}{3})^{x-y} = \frac{4}{9} = (\frac{2}{3})^2

, which yields

x - y = 2

. Solving the system of equations

x + y = 4

and

x - y = 2

gives

x = 3

and

y = 1

. The value of

x^2 + y^2

is therefore

3^2 + 1^2 = 10$.

Step-by-Step Solution

Multiply the two equations together.

6^{x+y} = 1296

, which simplifies to

x + y = 4

Multiplying the equations groups bases of 2 and 3 together to form base 6, allowing us to find the sum of the variables.

Divide the first equation by the second equation.

(\frac{2}{3})^{x-y} = \frac{4}{9}

, which simplifies to

x - y = 2

Dividing the equations groups bases of 2 and 3 to form base

2/3

, allowing us to find the difference of the variables.

Solve the system of equations for

x

and

y

x = 3

and

y = 1

Solving the linear system of equations

x + y = 4

and

x - y = 2

gives the individual values of

x

and

y

Calculate

x^2 + y^2

10

Substitute the values of

x

and

y

into the target expression.

Key Concept

Solving systems of exponential equations using properties of exponents and bases

Alternative Method

Take the logarithm of both sides of each equation to convert them into a system of linear equations in terms of

x

and

y

x \log 2 + y \log 3 = \log 24

and

x \log 3 + y \log 2 = \log 54

. Solving this system using elimination or substitution yields

x = 3

and

y = 1

, so

x^2 + y^2 = 10

Estimated Time:2m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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