If x and y satisfy the system of equations below, what is the value of x + y? \begin{align… — SAT

If $x$ and $y$ satisfy the system of equations below, what is the value of $x + y$ ?

\begin{aligned} 8^{x-y} &= 2^{y+3} \\ 9^{x} &= 27^{y-1} \end{aligned}

Answer: 36

The correct answer is 36. By rewriting

8^{x-y} = 2^{y+3}

with base 2, we obtain the relation

3x - 4y = 3

. Similarly, rewriting

9^x = 27^{y-1}

with base 3 gives

2x - 3y = -3

. Solving this system of equations simultaneously yields

x = 21

and

y = 15

. Adding these values together gives the final result of 36.

Rewrite the first equation using the common base of 2.

2^{3(x-y)} = 2^{y+3}

, which simplifies to

3x - 4y = 3

Since

8 = 2^3

, we can apply the power of a power rule

(a^m)^n = a^{mn}

to rewrite

8^{x-y}

2^{3(x-y)}

. Setting the exponents equal gives the linear equation

3(x-y) = y+3

, which simplifies to

3x - 4y = 3

Rewrite the second equation using the common base of 3.

3^{2x} = 3^{3(y-1)}

, which simplifies to

2x - 3y = -3

Since

9 = 3^2

and

27 = 3^3

, we rewrite the terms using base 3. Equating the exponents gives

2x = 3(y-1)

, which simplifies to

2x - 3y = -3

Solve the system of two linear equations.

x = 21

and

y = 15

Using the elimination method, we can multiply the first equation by 3 and the second equation by 4 to align the coefficients of

y

, obtaining

9x - 12y = 9

and

8x - 12y = -12

. Subtracting the equations gives

x = 21

. Substituting this back yields

y = 15

Find the sum of

x

and

y

The question asks for the value of

x + y

. Substituting

x = 21

and

y = 15

gives

21 + 15 = 36

Solving systems of exponential equations by expressing bases as powers of common bases to establish a system of linear equations.

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