If \frac{2x - 3}{4} - \frac{x - 5}{3} = \frac{x + 15}{12}, what is the value of 3x - 2? — GRE General Test

If $\frac{2x - 3}{4} - \frac{x - 5}{3} = \frac{x + 15}{12}$ , what is the value of $3x - 2$ ?

The value of

3x - 2

10

Multiplying the given linear equation by the least common denominator 12 yields

3(2x - 3) - 4(x - 5) = x + 15

. Expanding both sides gives

6x - 9 - 4x + 20 = x + 15

, which simplifies to

2x + 11 = x + 15

, leading to

x = 4

. Substituting

x = 4

into

3x - 2

gives

3(4) - 2 = 10

Clear the denominators by multiplying both sides of the equation by the least common multiple of 4, 3, and 12, which is 12.

12 \cdot \left(\frac{2x - 3}{4}\right) - 12 \cdot \left(\frac{x - 5}{3}\right) = 12 \cdot \left(\frac{x + 15}{12}\right) \implies 3(2x - 3) - 4(x - 5) = x + 15

Eliminating fractions simplifies the algebraic manipulation.

Distribute the coefficients through the terms in parentheses, paying careful attention to the negative sign on the second term.

6x - 9 - 4x + 20 = x + 15

Distributing

-4

across

(x - 5)

yields

-4x + 20

Combine like terms on the left side of the equation.

2x + 11 = x + 15

Grouping variable terms

(6x - 4x = 2x)

and constant terms

(-9 + 20 = 11)

Isolate the variable

x

by subtracting

x

and

11

from both sides.

x = 4

Solves the linear equation for

x

Substitute

x = 4

into the target expression

3x - 2

3(4) - 2 = 12 - 2 = 10

Calculates the final answer requested in the stem.

Linear Equations with Fractional Coefficients

Question