Solve for a 1/3(4a+1)=1/2a

$\frac{1}{3} (4 a + 1) = \frac{1}{2} a$

Step 1

Simplify $\frac{1}{3} \cdot (4 a + 1)$ .

Start simplifying.

$0 + 0 + \frac{1}{3} \cdot (4 a + 1) = \frac{1}{2} a$

Simplify by adding zeros.

$\frac{1}{3} \cdot (4 a + 1) = \frac{1}{2} a$

Apply the distributive property.

$\frac{1}{3} (4 a) + \frac{1}{3} \cdot 1 = \frac{1}{2} a$

Multiply $\frac{1}{3} (4 a)$ .

Combine $4$ and $\frac{1}{3}$ .

$\frac{4}{3} a + \frac{1}{3} \cdot 1 = \frac{1}{2} a$

Combine $\frac{4}{3}$ and $a$ .

$\frac{4 a}{3} + \frac{1}{3} \cdot 1 = \frac{1}{2} a$

Multiply $\frac{1}{3}$ by $1$ .

$\frac{4 a}{3} + \frac{1}{3} = \frac{1}{2} a$

Step 2

Combine $\frac{1}{2}$ and $a$ .

$\frac{4 a}{3} + \frac{1}{3} = \frac{a}{2}$

Step 3

Move all terms containing $a$ to the left side of the equation.

Subtract $\frac{a}{2}$ from both sides of the equation.

$\frac{4 a}{3} + \frac{1}{3} - \frac{a}{2} = 0$

To write $\frac{4 a}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 a}{3} \cdot \frac{2}{2} - \frac{a}{2} + \frac{1}{3} = 0$

To write $- \frac{a}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 a}{3} \cdot \frac{2}{2} - \frac{a}{2} \cdot \frac{3}{3} + \frac{1}{3} = 0$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 a}{3}$ by $\frac{2}{2}$ .

$\frac{4 a \cdot 2}{3 \cdot 2} - \frac{a}{2} \cdot \frac{3}{3} + \frac{1}{3} = 0$

Multiply $3$ by $2$ .

$\frac{4 a \cdot 2}{6} - \frac{a}{2} \cdot \frac{3}{3} + \frac{1}{3} = 0$

Multiply $\frac{a}{2}$ by $\frac{3}{3}$ .

$\frac{4 a \cdot 2}{6} - \frac{a \cdot 3}{2 \cdot 3} + \frac{1}{3} = 0$

Multiply $2$ by $3$ .

$\frac{4 a \cdot 2}{6} - \frac{a \cdot 3}{6} + \frac{1}{3} = 0$

Combine the numerators over the common denominator.

$\frac{4 a \cdot 2 - a \cdot 3}{6} + \frac{1}{3} = 0$

Simplify each term.

Simplify the numerator.

Factor $a$ out of $4 a \cdot 2 - a \cdot 3$ .

Factor $a$ out of $4 a \cdot 2$ .

$\frac{a (4 \cdot 2) - a \cdot 3}{6} + \frac{1}{3} = 0$

Factor $a$ out of $- a \cdot 3$ .

$\frac{a (4 \cdot 2) + a (- 1 \cdot 3)}{6} + \frac{1}{3} = 0$

Factor $a$ out of $a (4 \cdot 2) + a (- 1 \cdot 3)$ .

$\frac{a (4 \cdot 2 - 1 \cdot 3)}{6} + \frac{1}{3} = 0$

Multiply $4$ by $2$ .

$\frac{a (8 - 1 \cdot 3)}{6} + \frac{1}{3} = 0$

Multiply $- 1$ by $3$ .

$\frac{a (8 - 3)}{6} + \frac{1}{3} = 0$

Subtract $3$ from $8$ .

$\frac{a \cdot 5}{6} + \frac{1}{3} = 0$

Move $5$ to the left of $a$ .

$\frac{5 a}{6} + \frac{1}{3} = 0$

Step 4

Subtract $\frac{1}{3}$ from both sides of the equation.

$\frac{5 a}{6} = - \frac{1}{3}$

Step 5

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 a}{6} = \frac{6}{5} (- \frac{1}{3})$

Step 6

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{6}{5} \cdot \frac{5 a}{6}$ .

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 a}{6} = \frac{6}{5} (- \frac{1}{3})$

Rewrite the expression.

$\frac{1}{5} (5 a) = \frac{6}{5} (- \frac{1}{3})$

Cancel the common factor of $5$ .

Factor $5$ out of $5 a$ .

$\frac{1}{5} (5 (a)) = \frac{6}{5} (- \frac{1}{3})$

Cancel the common factor.

$\frac{1}{5} (5 a) = \frac{6}{5} (- \frac{1}{3})$

Rewrite the expression.

$a = \frac{6}{5} (- \frac{1}{3})$

Simplify the right side.

Simplify $\frac{6}{5} (- \frac{1}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$a = \frac{6}{5} \cdot \frac{- 1}{3}$

Factor $3$ out of $6$ .

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

Cancel the common factor.

$a = \frac{3 \cdot 2}{5} \cdot \frac{- 1}{3}$

Rewrite the expression.

$a = \frac{2}{5} \cdot - 1$

Combine $\frac{2}{5}$ and $- 1$ .

$a = \frac{2 \cdot - 1}{5}$

Simplify the expression.

Multiply $2$ by $- 1$ .

$a = \frac{- 2}{5}$

Move the negative in front of the fraction.

$a = - \frac{2}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{2}{5}$

Decimal Form:

$a = - 0.4$

Solve for a 1/3(4a+1)=1/2a

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