If 3^{2x - 1} \cdot 9^{y + 1} = 27^{x + y}, which of the following equations expresses x i… — SAT

If $3^{2x - 1} \cdot 9^{y + 1} = 27^{x + y}$ , which of the following equations expresses $x$ in terms of $y$ ?

The equation expressing

x

in terms of

y

x = 1 - y

The correct option is

x = 1 - y

. To solve the equation, each term is rewritten with base 3:

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

. Using the exponent rule

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

, this becomes

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

. Using the rule

b^m \cdot b^n = b^{m+n}

, the left side simplifies to

3^{2x+2y+1}

. Equating the exponents gives

2x+2y+1 = 3x+3y

. Subtracting

2x

and

2y

from both sides results in

1 = x+y

, which isolated for

x

gives

x = 1-y

Rewrite all bases in the equation as powers of 3.

Since

9 = 3^2

and

27 = 3^3

, the equation becomes

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

To apply exponent rules and combine terms, all bases must be identical.

Apply the power of a power rule

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

to simplify the exponents.

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

Multiplying the outer exponent by the inner exponent simplifies the terms on both sides.

Apply the product of powers rule

b^m \cdot b^n = b^{m+n}

to combine the terms on the left side of the equation.

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

, which simplifies to

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

Adding exponents when multiplying expressions with the same base allows the left side to be represented as a single exponential term.

Set the exponents equal to each other and solve for

x

in terms of

y

2x + 2y + 1 = 3x + 3y \implies 1 = x + y \implies x = 1 - y

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must be equal.

Solving exponential equations by expressing terms with a common base and applying exponent laws.

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