If 27^{2x - 1} \cdot 9^{y + 2} = 3^{13} and 8^{x - y} = 2^{y - 9}, what is the value of x … — SAT

If $27^{2x - 1} \cdot 9^{y + 2} = 3^{13}$ and $8^{x - y} = 2^{y - 9}$ , what is the value of $x + y$ ?

To solve the system, we convert all exponential terms to their prime bases. The first equation becomes

3^{6x - 3} \cdot 3^{2y + 4} = 3^{13}

, which simplifies to

3^{6x + 2y + 1} = 3^{13}

. Equating the exponents yields

6x + 2y + 1 = 13

, which simplifies to

3x + y = 6

. The second equation becomes

2^{3x - 3y} = 2^{y - 9}

. Equating these exponents yields

3x - 4y = -9

. Solving this system of equations gives

x = 1

and

y = 3

. The sum of these values is

1 + 3 = 4

Convert the bases of the first equation to a common base of 3.

Since

27 = 3^3

and

9 = 3^2

, we can rewrite the equation

27^{2x - 1} \cdot 9^{y + 2} = 3^{13}

(3^3)^{2x - 1} \cdot (3^2)^{y + 2} = 3^{13}

. Applying the power of a power rule, this simplifies to

3^{6x - 3} \cdot 3^{2y + 4} = 3^{13}

. Using the product rule of exponents, we add the exponents to get

3^{6x + 2y + 1} = 3^{13}

This allows us to equate the exponents and form a linear equation.

Set the exponents equal to each other to obtain the first linear equation.

6x + 2y + 1 = 13 \implies 6x + 2y = 12 \implies 3x + y = 6

Since the bases are identical on both sides, their exponents must be equal.

Convert the bases of the second equation to a common base of 2 and equate their exponents.

Since

8 = 2^3

, rewrite

8^{x - y} = 2^{y - 9}

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

. Equating the exponents gives

3x - 3y = y - 9 \implies 3x - 4y = -9

This gives us a second linear equation to form a system of equations.

Solve the system of equations for the variables.

From the first equation, we have

3x = 6 - y

. Substituting this into the second equation gives

(6 - y) - 4y = -9 \implies 6 - 5y = -9 \implies -5y = -15 \implies y = 3

. Substituting

y = 3

back into the first equation yields

3x + 3 = 6 \implies 3x = 3 \implies x = 1

Solving the system of linear equations provides the individual values of the variables.

Calculate the sum of the variables.

x + y = 1 + 3 = 4

This answers the question asking for the sum of the variables.

Expressing exponential terms with a common base to form and solve a system of linear equations.

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