If x satisfies the linear equation \frac{3(x - 2)}{4} - \frac{x - 5}{3} = \frac{2x + 1}{6}… — GRE General Test

x

satisfies the linear equation

\frac{3(x - 2)}{4} - \frac{x - 5}{3} = \frac{2x + 1}{6} + 2

what is the value of

2x - 5

The correct answer is obtained by clearing denominators with the least common denominator 12, yielding

9(x - 2) - 4(x - 5) = 2(2x + 1) + 24

. Expanding and simplifying gives

5x + 2 = 4x + 26

, which isolates

x = 24

. Substituting

x = 24

into

2x - 5

results in

2(24) - 5 = 43

Clear the denominators by multiplying every term in the equation by the least common multiple of 4, 3, and 6, which is 12.

12 \cdot \frac{3(x - 2)}{4} - 12 \cdot \frac{x - 5}{3} = 12 \cdot \frac{2x + 1}{6} + 12 \cdot 2 \implies 9(x - 2) - 4(x - 5) = 2(2x + 1) + 24

Eliminating fractions simplifies the linear equation into standard polynomial form.

Distribute the constants through the parentheses, taking careful note of negative signs.

9x - 18 - 4x + 20 = 4x + 2 + 24

Parentheses must be removed to collect like variable and constant terms.

Combine like terms on both sides of the equation.

5x + 2 = 4x + 26

Consolidating terms on each side allows for isolating the variable.

Isolate

x

on one side of the equation.

5x - 4x = 26 - 2 \implies x = 24

Subtracting

4x

and 2 from both sides solves directly for

x

Evaluate the target expression

2x - 5

using

x = 24

2(24) - 5 = 48 - 5 = 43

The question requests the value of

2x - 5

, not

x

alone.

Solving multi-step linear equations in one variable with fractional terms and evaluating algebraic expressions.

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