To solve for \( x \) in the equation
\[ \frac{27}{2x - 5} + 1 = 2, \]
we can start by isolating the fraction on one side.
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Subtract 1 from both sides: \[ \frac{27}{2x - 5} = 2 - 1, \] which simplifies to \[ \frac{27}{2x - 5} = 1. \]
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Multiply both sides by \( 2x - 5 \) (assuming \( 2x - 5 \neq 0 \)): \[ 27 = 1(2x - 5). \]
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Distribute on the right: \[ 27 = 2x - 5. \]
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Add 5 to both sides: \[ 27 + 5 = 2x, \] which simplifies to \[ 32 = 2x. \]
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Divide both sides by 2: \[ x = \frac{32}{2} = 16. \]
Thus, the solution is \( x = 16 \).
Explanation of why the answer is 16:
- We correctly manipulated the equation to isolate \( x \).
- We ensured each step followed algebraic principles, leading us step-by-step to the solution.
- Substituting \( x = 16 \) back into the original equation verifies the solution:
\[ 2x - 5 = 2(16) - 5 = 32 - 5 = 27, \]
and
\[ \frac{27}{27} + 1 = 1 + 1 = 2, \] which matches the right-hand side of the original equation.
Thus, the value \( x = 16 \) satisfies the equation, confirming it is correct.
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