If 9^x + 1 = 27^x, what is the value of x? | SAT Nonlinear equations in one variable and systems of equations in two variables Question, Answer Explained

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Math · Advanced MathMedium

If $9^{x + 1} = 27^x$ , what is the value of $x$ ?

$\frac{1}{2}$

$1$

$2$

$3$

Show answer and explanation

Answer: C

Write both sides as powers of $3$ : $9^{x + 1} = (3^2)^{x + 1} = 3^{2x + 2}$ and $27^x = (3^3)^x = 3^{3x}$ . Setting the exponents equal gives $2x + 2 = 3x$ , so $x = 2$ . Check: $9^3 = 729$ and $27^2 = 729$ .

Choice B multiplies only the $x$ by $2$ , writing $3^{2x + 1}$ instead of $3^{2x + 2}$ .

Why the other choices are wrong

A. This sets x + 1 equal to 3x, mixing an exponent of base 9 with an exponent of base 3. Write 9 as 3² first.
B. This multiplies only the x by 2. The exponent of 3 on the left is 2(x + 1) = 2x + 2.
D. Substituting 3 gives 9⁴ = 6,561 and 27³ = 19,683, which aren't equal.

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