If \frac{1}{2}(4x - 8) - \frac{2}{3}(3 - 9x) = 5(x - 1) + 7, what is the value of 3x - 2? — SAT

If $\frac{1}{2}(4x - 8) - \frac{2}{3}(3 - 9x) = 5(x - 1) + 7$ , what is the value of $3x - 2$ ?

The correct value is 6. After correctly distributing the coefficients on both sides, the equation becomes

8x - 6 = 5x + 2

. Isolating the variable yields

3x = 8

, which gives

x = \frac{8}{3}

. Substituting this back into the target expression

3x - 2

results in

3(\frac{8}{3}) - 2 = 6

Distribute the fraction coefficients on the left side of the equation.

\frac{1}{2}(4x - 8) - \frac{2}{3}(3 - 9x) = 2x - 4 - 2 + 6x = 8x - 6

To simplify the expression by removing the parentheses on the left side.

Distribute and simplify the right side of the equation.

5(x - 1) + 7 = 5x - 5 + 7 = 5x + 2

To simplify the expression by removing the parentheses on the right side.

Equate the simplified left and right sides, then solve for

x

8x - 6 = 5x + 2 \implies 3x = 8 \implies x = \frac{8}{3}

To isolate the variable

x

Substitute the value of

x

into the expression

3x - 2

3\left(\frac{8}{3}\right) - 2 = 8 - 2 = 6

To find the final value requested by the question.

Solving linear equations in one variable by distributing coefficients, combining like terms, and isolating the variable.

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