Solve for a 19/4-3 1/2a=-1/4a+3/2

$\frac{19}{4} - 3 \frac{1}{2} a = - \frac{1}{4} a + \frac{3}{2}$

Simplify the left side.

Simplify $\frac{19}{4} - 3 \frac{1}{2} a$ .

Convert $3 \frac{1}{2}$ to an improper fraction.

A mixed number is an addition of its whole and fractional parts.

$\frac{19}{4} - (3 + \frac{1}{2}) a = - \frac{1}{4} a + \frac{3}{2}$

Add $3$ and $\frac{1}{2}$ .

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

$\frac{19}{4} - (3 \cdot \frac{2}{2} + \frac{1}{2}) a = - \frac{1}{4} a + \frac{3}{2}$

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

$\frac{19}{4} - (\frac{3 \cdot 2}{2} + \frac{1}{2}) a = - \frac{1}{4} a + \frac{3}{2}$

Combine the numerators over the common denominator.

$\frac{19}{4} - \frac{3 \cdot 2 + 1}{2} a = - \frac{1}{4} a + \frac{3}{2}$

Simplify the numerator.

Multiply $3$ by $2$ .

$\frac{19}{4} - \frac{6 + 1}{2} a = - \frac{1}{4} a + \frac{3}{2}$

Add $6$ and $1$ .

$\frac{19}{4} - \frac{7}{2} a = - \frac{1}{4} a + \frac{3}{2}$

Simplify each term.

Combine $a$ and $\frac{7}{2}$ .

$\frac{19}{4} - \frac{a \cdot 7}{2} = - \frac{1}{4} a + \frac{3}{2}$

Move $7$ to the left of $a$ .

$\frac{19}{4} - \frac{7 a}{2} = - \frac{1}{4} a + \frac{3}{2}$

Simplify the right side.

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

$\frac{19}{4} - \frac{7 a}{2} = - \frac{a}{4} + \frac{3}{2}$

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

Add $\frac{a}{4}$ to both sides of the equation.

$\frac{19}{4} - \frac{7 a}{2} + \frac{a}{4} = \frac{3}{2}$

To write $- \frac{7 a}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{19}{4} - \frac{7 a}{2} \cdot \frac{2}{2} + \frac{a}{4} = \frac{3}{2}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7 a}{2}$ by $\frac{2}{2}$ .

$\frac{19}{4} - \frac{7 a \cdot 2}{2 \cdot 2} + \frac{a}{4} = \frac{3}{2}$

Multiply $2$ by $2$ .

$\frac{19}{4} - \frac{7 a \cdot 2}{4} + \frac{a}{4} = \frac{3}{2}$

Combine the numerators over the common denominator.

$\frac{19}{4} + \frac{- 7 a \cdot 2 + a}{4} = \frac{3}{2}$

Combine the numerators over the common denominator.

$\frac{19 - 7 a \cdot 2 + a}{4} = \frac{3}{2}$

Multiply $2$ by $- 7$ .

$\frac{19 - 14 a + a}{4} = \frac{3}{2}$

Add $- 14 a$ and $a$ .

$\frac{19 - 13 a}{4} = \frac{3}{2}$

Multiply both sides by $4$ .

$\frac{19 - 13 a}{4} \cdot 4 = \frac{3}{2} \cdot 4$

Simplify.

Simplify the left side.

Simplify $\frac{19 - 13 a}{4} \cdot 4$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{19 - 13 a}{4} \cdot 4 = \frac{3}{2} \cdot 4$

Rewrite the expression.

$19 - 13 a = \frac{3}{2} \cdot 4$

Reorder $19$ and $- 13 a$ .

$- 13 a + 19 = \frac{3}{2} \cdot 4$

Simplify the right side.

Simplify $\frac{3}{2} \cdot 4$ .

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$- 13 a + 19 = \frac{3}{2} \cdot (2 (2))$

Cancel the common factor.

$- 13 a + 19 = \frac{3}{2} \cdot (2 \cdot 2)$

Rewrite the expression.

$- 13 a + 19 = 3 \cdot 2$

Multiply $3$ by $2$ .

$- 13 a + 19 = 6$

Solve for $a$ .

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

Subtract $19$ from both sides of the equation.

$- 13 a = 6 - 19$

Subtract $19$ from $6$ .

$- 13 a = - 13$

Divide each term in $- 13 a = - 13$ by $- 13$ and simplify.

Divide each term in $- 13 a = - 13$ by $- 13$ .

$\frac{- 13 a}{- 13} = \frac{- 13}{- 13}$

Simplify the left side.

Cancel the common factor of $- 13$ .

Cancel the common factor.

$\frac{- 13 a}{- 13} = \frac{- 13}{- 13}$

Divide $a$ by $1$ .

$a = \frac{- 13}{- 13}$

Simplify the right side.

Divide $- 13$ by $- 13$ .

$a = 1$

Solve for a 19/4-3 1/2a=-1/4a+3/2

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