Finite Math Examples

$\frac{1}{9} \cdot (a + 7) = \frac{1}{4} \cdot (7 - a)$

Step 1

Simplify $\frac{1}{9} \cdot (a + 7)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{9} \cdot (a + 7) = \frac{1}{4} \cdot (7 - a)$

Step 1.2

Simplify by adding zeros.

$\frac{1}{9} \cdot (a + 7) = \frac{1}{4} \cdot (7 - a)$

Step 1.3

Apply the distributive property.

$\frac{1}{9} a + \frac{1}{9} \cdot 7 = \frac{1}{4} \cdot (7 - a)$

Step 1.4

Combine $\frac{1}{9}$ and $a$ .

$\frac{a}{9} + \frac{1}{9} \cdot 7 = \frac{1}{4} \cdot (7 - a)$

Step 1.5

Combine $\frac{1}{9}$ and $7$ .

$\frac{a}{9} + \frac{7}{9} = \frac{1}{4} \cdot (7 - a)$

Step 2

Simplify $\frac{1}{4} \cdot (7 - a)$ .

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Step 2.1

Apply the distributive property.

$\frac{a}{9} + \frac{7}{9} = \frac{1}{4} \cdot 7 + \frac{1}{4} (- a)$

Step 2.2

Combine $\frac{1}{4}$ and $7$ .

$\frac{a}{9} + \frac{7}{9} = \frac{7}{4} + \frac{1}{4} (- a)$

Step 2.3

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

$\frac{a}{9} + \frac{7}{9} = \frac{7}{4} - \frac{a}{4}$

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.

$\frac{a}{9} + \frac{7}{9} + \frac{a}{4} = \frac{7}{4}$

Step 3.2

To write $\frac{a}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{a}{9} \cdot \frac{4}{4} + \frac{a}{4} + \frac{7}{9} = \frac{7}{4}$

Step 3.3

To write $\frac{a}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{a}{9} \cdot \frac{4}{4} + \frac{a}{4} \cdot \frac{9}{9} + \frac{7}{9} = \frac{7}{4}$

Step 3.4

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{a}{9}$ by $\frac{4}{4}$ .

$\frac{a \cdot 4}{9 \cdot 4} + \frac{a}{4} \cdot \frac{9}{9} + \frac{7}{9} = \frac{7}{4}$

Step 3.4.2

Multiply $9$ by $4$ .

$\frac{a \cdot 4}{36} + \frac{a}{4} \cdot \frac{9}{9} + \frac{7}{9} = \frac{7}{4}$

Step 3.4.3

Multiply $\frac{a}{4}$ by $\frac{9}{9}$ .

$\frac{a \cdot 4}{36} + \frac{a \cdot 9}{4 \cdot 9} + \frac{7}{9} = \frac{7}{4}$

Step 3.4.4

Multiply $4$ by $9$ .

$\frac{a \cdot 4}{36} + \frac{a \cdot 9}{36} + \frac{7}{9} = \frac{7}{4}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{a \cdot 4 + a \cdot 9}{36} + \frac{7}{9} = \frac{7}{4}$

Step 3.6

Reorder $a$ and $9$ .

$\frac{4 \cdot a + 9 \cdot a}{36} + \frac{7}{9} = \frac{7}{4}$

Step 3.7

Add $4 \cdot a$ and $9 \cdot a$ .

$\frac{13 a}{36} + \frac{7}{9} = \frac{7}{4}$

Step 4

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

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Step 4.1

Subtract $\frac{7}{9}$ from both sides of the equation.

$\frac{13 a}{36} = \frac{7}{4} - \frac{7}{9}$

Step 4.2

To write $\frac{7}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{13 a}{36} = \frac{7}{4} \cdot \frac{9}{9} - \frac{7}{9}$

Step 4.3

To write $- \frac{7}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{13 a}{36} = \frac{7}{4} \cdot \frac{9}{9} - \frac{7}{9} \cdot \frac{4}{4}$

Step 4.4

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{7}{4}$ by $\frac{9}{9}$ .

$\frac{13 a}{36} = \frac{7 \cdot 9}{4 \cdot 9} - \frac{7}{9} \cdot \frac{4}{4}$

Step 4.4.2

Multiply $4$ by $9$ .

$\frac{13 a}{36} = \frac{7 \cdot 9}{36} - \frac{7}{9} \cdot \frac{4}{4}$

Step 4.4.3

Multiply $\frac{7}{9}$ by $\frac{4}{4}$ .

$\frac{13 a}{36} = \frac{7 \cdot 9}{36} - \frac{7 \cdot 4}{9 \cdot 4}$

Step 4.4.4

Multiply $9$ by $4$ .

$\frac{13 a}{36} = \frac{7 \cdot 9}{36} - \frac{7 \cdot 4}{36}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{13 a}{36} = \frac{7 \cdot 9 - 7 \cdot 4}{36}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $7$ by $9$ .

$\frac{13 a}{36} = \frac{63 - 7 \cdot 4}{36}$

Step 4.6.2

Multiply $- 7$ by $4$ .

$\frac{13 a}{36} = \frac{63 - 28}{36}$

Step 4.6.3

Subtract $28$ from $63$ .

$\frac{13 a}{36} = \frac{35}{36}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$13 a = 35$

Step 6

Divide each term in $13 a = 35$ by $13$ and simplify.

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Step 6.1

Divide each term in $13 a = 35$ by $13$ .

$\frac{13 a}{13} = \frac{35}{13}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $13$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{13 a}{13} = \frac{35}{13}$

Step 6.2.1.2

Divide $a$ by $1$ .

$a = \frac{35}{13}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{35}{13}$

Decimal Form:

$a = 2.\overline{692307}$

Mixed Number Form:

$a = 2 \frac{9}{13}$