Solve for a 9/8(4-1/2a)=3(1/4a+5/2)

$\frac{9}{8} (4 - \frac{1}{2} a) = 3 (\frac{1}{4} a + \frac{5}{2})$

Step 1

Simplify $\frac{9}{8} \cdot (4 - \frac{1}{2} a)$ .

Start simplifying.

$0 + 0 + \frac{9}{8} \cdot (4 - \frac{1}{2} a) = 3 (\frac{1}{4} a + \frac{5}{2})$

Simplify by adding zeros.

$\frac{9}{8} \cdot (4 - \frac{1}{2} a) = 3 (\frac{1}{4} a + \frac{5}{2})$

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

$\frac{9}{8} \cdot (4 - \frac{a}{2}) = 3 (\frac{1}{4} a + \frac{5}{2})$

Apply the distributive property.

$\frac{9}{8} \cdot 4 + \frac{9}{8} (- \frac{a}{2}) = 3 (\frac{1}{4} a + \frac{5}{2})$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$\frac{9}{4 (2)} \cdot 4 + \frac{9}{8} (- \frac{a}{2}) = 3 (\frac{1}{4} a + \frac{5}{2})$

Cancel the common factor.

$\frac{9}{4 \cdot 2} \cdot 4 + \frac{9}{8} (- \frac{a}{2}) = 3 (\frac{1}{4} a + \frac{5}{2})$

Rewrite the expression.

$\frac{9}{2} + \frac{9}{8} (- \frac{a}{2}) = 3 (\frac{1}{4} a + \frac{5}{2})$

Multiply $\frac{9}{8} (- \frac{a}{2})$ .

Multiply $\frac{9}{8}$ by $\frac{a}{2}$ .

$\frac{9}{2} - \frac{9 a}{8 \cdot 2} = 3 (\frac{1}{4} a + \frac{5}{2})$

Multiply $8$ by $2$ .

$\frac{9}{2} - \frac{9 a}{16} = 3 (\frac{1}{4} a + \frac{5}{2})$

Step 2

Simplify $3 (\frac{1}{4} a + \frac{5}{2})$ .

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

$\frac{9}{2} - \frac{9 a}{16} = 3 (\frac{a}{4} + \frac{5}{2})$

Apply the distributive property.

$\frac{9}{2} - \frac{9 a}{16} = 3 \frac{a}{4} + 3 (\frac{5}{2})$

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

$\frac{9}{2} - \frac{9 a}{16} = \frac{3 a}{4} + 3 (\frac{5}{2})$

Multiply $3 (\frac{5}{2})$ .

Combine $3$ and $\frac{5}{2}$ .

$\frac{9}{2} - \frac{9 a}{16} = \frac{3 a}{4} + \frac{3 \cdot 5}{2}$

Multiply $3$ by $5$ .

$\frac{9}{2} - \frac{9 a}{16} = \frac{3 a}{4} + \frac{15}{2}$

Step 3

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

Subtract $\frac{3 a}{4}$ from both sides of the equation.

$\frac{9}{2} - \frac{9 a}{16} - \frac{3 a}{4} = \frac{15}{2}$

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

$\frac{9}{2} - \frac{9 a}{16} - \frac{3 a}{4} \cdot \frac{4}{4} = \frac{15}{2}$

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{9}{2} - \frac{9 a}{16} - \frac{3 a \cdot 4}{4 \cdot 4} = \frac{15}{2}$

Multiply $4$ by $4$ .

$\frac{9}{2} - \frac{9 a}{16} - \frac{3 a \cdot 4}{16} = \frac{15}{2}$

Combine the numerators over the common denominator.

$\frac{9}{2} + \frac{- 9 a - 3 a \cdot 4}{16} = \frac{15}{2}$

Simplify each term.

Simplify the numerator.

Factor $3 a$ out of $- 9 a - 3 a \cdot 4$ .

Factor $3 a$ out of $- 9 a$ .

$\frac{9}{2} + \frac{3 a (- 3) - 3 a \cdot 4}{16} = \frac{15}{2}$

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

$\frac{9}{2} + \frac{3 a (- 3) + 3 a (- 1 \cdot 4)}{16} = \frac{15}{2}$

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

$\frac{9}{2} + \frac{3 a (- 3 - 1 \cdot 4)}{16} = \frac{15}{2}$

Multiply $- 1$ by $4$ .

$\frac{9}{2} + \frac{3 a (- 3 - 4)}{16} = \frac{15}{2}$

Subtract $4$ from $- 3$ .

$\frac{9}{2} + \frac{3 a \cdot - 7}{16} = \frac{15}{2}$

Multiply $- 7$ by $3$ .

$\frac{9}{2} + \frac{- 21 a}{16} = \frac{15}{2}$

Move the negative in front of the fraction.

$\frac{9}{2} - \frac{21 a}{16} = \frac{15}{2}$

Step 4

Move all terms not containing $a$ to the right side of the equation.

Subtract $\frac{9}{2}$ from both sides of the equation.

$- \frac{21 a}{16} = \frac{15}{2} - \frac{9}{2}$

Combine the numerators over the common denominator.

$- \frac{21 a}{16} = \frac{15 - 9}{2}$

Subtract $9$ from $15$ .

$- \frac{21 a}{16} = \frac{6}{2}$

Divide $6$ by $2$ .

$- \frac{21 a}{16} = 3$

Step 5

Multiply both sides of the equation by $- \frac{16}{21}$ .

$- \frac{16}{21} (- \frac{21 a}{16}) = - \frac{16}{21} \cdot 3$

Step 6

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{16}{21} (- \frac{21 a}{16})$ .

Cancel the common factor of $16$ .

Move the leading negative in $- \frac{16}{21}$ into the numerator.

$\frac{- 16}{21} (- \frac{21 a}{16}) = - \frac{16}{21} \cdot 3$

Move the leading negative in $- \frac{21 a}{16}$ into the numerator.

$\frac{- 16}{21} \cdot \frac{- 21 a}{16} = - \frac{16}{21} \cdot 3$

Factor $16$ out of $- 16$ .

$\frac{16 (- 1)}{21} \cdot \frac{- 21 a}{16} = - \frac{16}{21} \cdot 3$

Cancel the common factor.

$\frac{16 \cdot - 1}{21} \cdot \frac{- 21 a}{16} = - \frac{16}{21} \cdot 3$

Rewrite the expression.

$\frac{- 1}{21} (- 21 a) = - \frac{16}{21} \cdot 3$

Cancel the common factor of $21$ .

Factor $21$ out of $- 21 a$ .

$\frac{- 1}{21} (21 (- a)) = - \frac{16}{21} \cdot 3$

Cancel the common factor.

$\frac{- 1}{21} (21 (- a)) = - \frac{16}{21} \cdot 3$

Rewrite the expression.

$- - a = - \frac{16}{21} \cdot 3$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 a = - \frac{16}{21} \cdot 3$

Multiply $a$ by $1$ .

$a = - \frac{16}{21} \cdot 3$

Simplify the right side.

Simplify $- \frac{16}{21} \cdot 3$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{16}{21}$ into the numerator.

$a = \frac{- 16}{21} \cdot 3$

Factor $3$ out of $21$ .

$a = \frac{- 16}{3 (7)} \cdot 3$

Cancel the common factor.

$a = \frac{- 16}{3 \cdot 7} \cdot 3$

Rewrite the expression.

$a = \frac{- 16}{7}$

Move the negative in front of the fraction.

$a = - \frac{16}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{16}{7}$

Decimal Form:

$a = - 2.\overline{285714}$

Mixed Number Form:

$a = - 2 \frac{2}{7}$

Solve for a 9/8(4-1/2a)=3(1/4a+5/2)

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