In the equation below, a is a constant. \frac{1}{3}(ax - 2) - \frac{1}{2}(x - 4a) = 3x - 5… — SAT

In the equation below, $a$ is a constant.

\frac{1}{3}(ax - 2) - \frac{1}{2}(x - 4a) = 3x - 5

If the solution to the equation is $x = 2$ , what is the value of $a$ ?

Substituting

x = 2

into the equation gives

\frac{1}{3}(2a - 2) - \frac{1}{2}(2 - 4a) = 3(2) - 5

, which simplifies to

\frac{1}{3}(2a - 2) - (1 - 2a) = 1

. Multiplying the entire equation by 3 to clear the fraction yields

(2a - 2) - 3(1 - 2a) = 3

. Distributing the negative 3 yields

2a - 2 - 3 + 6a = 3

, which simplifies to

8a - 5 = 3

. Adding 5 to both sides results in

8a = 8

, which gives

a = 1

Substitute

x = 2

into the given equation.

\frac{1}{3}(2a - 2) - \frac{1}{2}(2 - 4a) = 3(2) - 5

Since

x = 2

is a solution to the equation, substituting it into the equation must yield a true mathematical statement.

Simplify the constant terms and distribute or simplify the expressions.

\frac{1}{3}(2a - 2) - (1 - 2a) = 1

Simplifying

3(2) - 5

yields

1

, and simplifying the second fraction term gives

\frac{1}{2}(2 - 4a) = 1 - 2a

Multiply the entire equation by 3 to eliminate the fraction.

(2a - 2) - 3(1 - 2a) = 3

Multiplying all terms on both sides of the equation by the denominator 3 clears the fraction and simplifies the equation.

Distribute the negative 3 and combine like terms to solve for

a

2a - 2 - 3 + 6a = 3 \implies 8a - 5 = 3 \implies 8a = 8 \implies a = 1

Distributing

-3(1 - 2a)

yields

-3 + 6a

. Combining like terms gives

8a - 5 = 3

, and adding 5 followed by dividing by 8 isolates the variable

a

Solving linear equations in one variable containing fractions and constant parameters by substitution and term isolation.

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