Basic Math Examples

$\frac{a - 6}{3} - 1 = \frac{a - 9}{21} + \frac{3 - a}{7}$

Step 1

Since $a$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{a - 9}{21} + \frac{3 - a}{7} = \frac{a - 6}{3} - 1$

Step 2

Simplify $\frac{a - 9}{21} + \frac{3 - a}{7}$ .

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Step 2.1

To write $\frac{3 - a}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{a - 9}{21} + \frac{3 - a}{7} \cdot \frac{3}{3} = \frac{a - 6}{3} - 1$

Step 2.2

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

Multiply $\frac{3 - a}{7}$ by $\frac{3}{3}$ .

$\frac{a - 9}{21} + \frac{(3 - a) \cdot 3}{7 \cdot 3} = \frac{a - 6}{3} - 1$

Step 2.2.2

Multiply $7$ by $3$ .

$\frac{a - 9}{21} + \frac{(3 - a) \cdot 3}{21} = \frac{a - 6}{3} - 1$

Step 2.3

Combine the numerators over the common denominator.

$\frac{a - 9 + (3 - a) \cdot 3}{21} = \frac{a - 6}{3} - 1$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{a - 9 + 3 \cdot 3 - a \cdot 3}{21} = \frac{a - 6}{3} - 1$

Step 2.4.2

Multiply $3$ by $3$ .

$\frac{a - 9 + 9 - a \cdot 3}{21} = \frac{a - 6}{3} - 1$

Step 2.4.3

Multiply $3$ by $- 1$ .

$\frac{a - 9 + 9 - 3 a}{21} = \frac{a - 6}{3} - 1$

Step 2.4.4

Subtract $3 a$ from $a$ .

$\frac{- 2 a - 9 + 9}{21} = \frac{a - 6}{3} - 1$

Step 2.4.5

Add $- 9$ and $9$ .

$\frac{- 2 a + 0}{21} = \frac{a - 6}{3} - 1$

Step 2.4.6

Add $- 2 a$ and $0$ .

$\frac{- 2 a}{21} = \frac{a - 6}{3} - 1$

Step 2.5

Move the negative in front of the fraction.

$- \frac{2 a}{21} = \frac{a - 6}{3} - 1$

Step 3

Simplify $\frac{a - 6}{3} - 1$ .

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Step 3.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2 a}{21} = \frac{a - 6}{3} - 1 \cdot \frac{3}{3}$

Step 3.2

Simplify terms.

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Step 3.2.1

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

$- \frac{2 a}{21} = \frac{a - 6}{3} + \frac{- 1 \cdot 3}{3}$

Step 3.2.2

Combine the numerators over the common denominator.

$- \frac{2 a}{21} = \frac{a - 6 - 1 \cdot 3}{3}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $- 1$ by $3$ .

$- \frac{2 a}{21} = \frac{a - 6 - 3}{3}$

Step 3.3.2

Subtract $3$ from $- 6$ .

$- \frac{2 a}{21} = \frac{a - 9}{3}$

Step 4

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

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Step 4.1

Subtract $\frac{a - 9}{3}$ from both sides of the equation.

$- \frac{2 a}{21} - \frac{a - 9}{3} = 0$

Step 4.2

To write $- \frac{a - 9}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{2 a}{21} - \frac{a - 9}{3} \cdot \frac{7}{7} = 0$

Step 4.3

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{a - 9}{3}$ by $\frac{7}{7}$ .

$- \frac{2 a}{21} - \frac{(a - 9) \cdot 7}{3 \cdot 7} = 0$

Step 4.3.2

Multiply $3$ by $7$ .

$- \frac{2 a}{21} - \frac{(a - 9) \cdot 7}{21} = 0$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 2 a - (a - 9) \cdot 7}{21} = 0$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Apply the distributive property.

$\frac{- 2 a + (- a - - 9) \cdot 7}{21} = 0$

Step 4.5.2

Multiply $- 1$ by $- 9$ .

$\frac{- 2 a + (- a + 9) \cdot 7}{21} = 0$

Step 4.5.3

Apply the distributive property.

$\frac{- 2 a - a \cdot 7 + 9 \cdot 7}{21} = 0$

Step 4.5.4

Multiply $7$ by $- 1$ .

$\frac{- 2 a - 7 a + 9 \cdot 7}{21} = 0$

Step 4.5.5

Multiply $9$ by $7$ .

$\frac{- 2 a - 7 a + 63}{21} = 0$

Step 4.5.6

Subtract $7 a$ from $- 2 a$ .

$\frac{- 9 a + 63}{21} = 0$

Step 4.5.7

Factor $9$ out of $- 9 a + 63$ .

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Step 4.5.7.1

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

$\frac{9 (- a) + 63}{21} = 0$

Step 4.5.7.2

Factor $9$ out of $63$ .

$\frac{9 (- a) + 9 (7)}{21} = 0$

Step 4.5.7.3

Factor $9$ out of $9 (- a) + 9 (7)$ .

$\frac{9 (- a + 7)}{21} = 0$

Step 4.6

Cancel the common factor of $9$ and $21$ .

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Step 4.6.1

Factor $3$ out of $9 (- a + 7)$ .

$\frac{3 (3 (- a + 7))}{21} = 0$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $3$ out of $21$ .

$\frac{3 (3 (- a + 7))}{3 (7)} = 0$

Step 4.6.2.2

Cancel the common factor.

$\frac{3 (3 (- a + 7))}{3 \cdot 7} = 0$

Step 4.6.2.3

Rewrite the expression.

$\frac{3 (- a + 7)}{7} = 0$

Step 4.7

Factor $- 1$ out of $- a$ .

$\frac{3 (- (a) + 7)}{7} = 0$

Step 4.8

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{3 (- (a) - 1 (- 7))}{7} = 0$

Step 4.9

Factor $- 1$ out of $- (a) - 1 (- 7)$ .

$\frac{3 (- (a - 7))}{7} = 0$

Step 4.10

Rewrite $- (a - 7)$ as $- 1 (a - 7)$ .

$\frac{3 (- 1 (a - 7))}{7} = 0$

Step 4.11

Move the negative in front of the fraction.

$- \frac{3 (a - 7)}{7} = 0$

Step 5

Set the numerator equal to zero.

$3 (a - 7) = 0$

Step 6

Solve the equation for $a$ .

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Step 6.1

Divide each term in $3 (a - 7) = 0$ by $3$ and simplify.

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Step 6.1.1

Divide each term in $3 (a - 7) = 0$ by $3$ .

$\frac{3 (a - 7)}{3} = \frac{0}{3}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $3$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{3 (a - 7)}{3} = \frac{0}{3}$

Step 6.1.2.1.2

Divide $a - 7$ by $1$ .

$a - 7 = \frac{0}{3}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $0$ by $3$ .

$a - 7 = 0$

Step 6.2

Add $7$ to both sides of the equation.

$a = 7$