Algebra Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

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

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

Combine the numerators over the common denominator.

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

Step 1.1.1.2.4

Simplify the numerator.

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Step 1.1.1.2.4.1

Multiply $3$ by $2$ .

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

Step 1.1.1.2.4.2

Add $6$ and $1$ .

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

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Multiply $2$ by $- 7$ .

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

Step 3.7

Add $- 14 a$ and $a$ .

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

Step 4

Multiply both sides by $4$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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Step 5.1.1.1

Cancel the common factor of $4$ .

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Step 5.1.1.1.1

Cancel the common factor.

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

Step 5.1.1.1.2

Rewrite the expression.

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

Step 5.1.1.2

Reorder $19$ and $- 13 a$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Factor $2$ out of $4$ .

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

Step 5.2.1.1.2

Cancel the common factor.

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

Step 5.2.1.1.3

Rewrite the expression.

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

Step 5.2.1.2

Multiply $3$ by $2$ .

$- 13 a + 19 = 6$

Step 6

Solve for $a$ .

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Step 6.1

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

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Step 6.1.1

Subtract $19$ from both sides of the equation.

$- 13 a = 6 - 19$

Step 6.1.2

Subtract $19$ from $6$ .

$- 13 a = - 13$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $- 13$ .

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

Divide $a$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $- 13$ by $- 13$ .

$a = 1$