Algebra Examples

$\frac{2}{3} (x - 3) - \frac{1}{7} (x + 1) = 1$

Step 1

Simplify $\frac{2}{3} \cdot (x - 3) - \frac{1}{7} \cdot (x + 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{2}{3} x + \frac{2}{3} \cdot - 3 - \frac{1}{7} \cdot (x + 1) = 1$

Step 1.1.2

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

$\frac{2 x}{3} + \frac{2}{3} \cdot - 3 - \frac{1}{7} \cdot (x + 1) = 1$

Step 1.1.3

Cancel the common factor of $3$ .

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Step 1.1.3.1

Factor $3$ out of $- 3$ .

$\frac{2 x}{3} + \frac{2}{3} \cdot (3 (- 1)) - \frac{1}{7} \cdot (x + 1) = 1$

Step 1.1.3.2

Cancel the common factor.

$\frac{2 x}{3} + \frac{2}{3} \cdot (3 \cdot - 1) - \frac{1}{7} \cdot (x + 1) = 1$

Step 1.1.3.3

Rewrite the expression.

$\frac{2 x}{3} + 2 \cdot - 1 - \frac{1}{7} \cdot (x + 1) = 1$

Step 1.1.4

Multiply $2$ by $- 1$ .

$\frac{2 x}{3} - 2 - \frac{1}{7} \cdot (x + 1) = 1$

Step 1.1.5

Apply the distributive property.

$\frac{2 x}{3} - 2 - \frac{1}{7} x - \frac{1}{7} \cdot 1 = 1$

Step 1.1.6

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

$\frac{2 x}{3} - 2 - \frac{x}{7} - \frac{1}{7} \cdot 1 = 1$

Step 1.1.7

Multiply $- 1$ by $1$ .

$\frac{2 x}{3} - 2 - \frac{x}{7} - \frac{1}{7} = 1$

Step 1.2

To write $\frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{2 x}{3} \cdot \frac{7}{7} - \frac{x}{7} - 2 - \frac{1}{7} = 1$

Step 1.3

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

$\frac{2 x}{3} \cdot \frac{7}{7} - \frac{x}{7} \cdot \frac{3}{3} - 2 - \frac{1}{7} = 1$

Step 1.4

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

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Step 1.4.1

Multiply $\frac{2 x}{3}$ by $\frac{7}{7}$ .

$\frac{2 x \cdot 7}{3 \cdot 7} - \frac{x}{7} \cdot \frac{3}{3} - 2 - \frac{1}{7} = 1$

Step 1.4.2

Multiply $3$ by $7$ .

$\frac{2 x \cdot 7}{21} - \frac{x}{7} \cdot \frac{3}{3} - 2 - \frac{1}{7} = 1$

Step 1.4.3

Multiply $\frac{x}{7}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 7}{21} - \frac{x \cdot 3}{7 \cdot 3} - 2 - \frac{1}{7} = 1$

Step 1.4.4

Multiply $7$ by $3$ .

$\frac{2 x \cdot 7}{21} - \frac{x \cdot 3}{21} - 2 - \frac{1}{7} = 1$

Step 1.5

Combine the numerators over the common denominator.

$\frac{2 x \cdot 7 - x \cdot 3}{21} - 2 - \frac{1}{7} = 1$

Step 1.6

Find the common denominator.

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Step 1.6.1

Write $- 2$ as a fraction with denominator $1$ .

$\frac{2 x \cdot 7 - x \cdot 3}{21} + \frac{- 2}{1} - \frac{1}{7} = 1$

Step 1.6.2

Multiply $\frac{- 2}{1}$ by $\frac{21}{21}$ .

$\frac{2 x \cdot 7 - x \cdot 3}{21} + \frac{- 2}{1} \cdot \frac{21}{21} - \frac{1}{7} = 1$

Step 1.6.3

Multiply $\frac{- 2}{1}$ by $\frac{21}{21}$ .

$\frac{2 x \cdot 7 - x \cdot 3}{21} + \frac{- 2 \cdot 21}{21} - \frac{1}{7} = 1$

Step 1.6.4

Multiply $\frac{1}{7}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 7 - x \cdot 3}{21} + \frac{- 2 \cdot 21}{21} - (\frac{1}{7} \cdot \frac{3}{3}) = 1$

Step 1.6.5

Multiply $\frac{1}{7}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 7 - x \cdot 3}{21} + \frac{- 2 \cdot 21}{21} - \frac{3}{7 \cdot 3} = 1$

Step 1.6.6

Reorder the factors of $7 \cdot 3$ .

$\frac{2 x \cdot 7 - x \cdot 3}{21} + \frac{- 2 \cdot 21}{21} - \frac{3}{3 \cdot 7} = 1$

Step 1.6.7

Multiply $3$ by $7$ .

$\frac{2 x \cdot 7 - x \cdot 3}{21} + \frac{- 2 \cdot 21}{21} - \frac{3}{21} = 1$

Step 1.7

Combine the numerators over the common denominator.

$\frac{2 x \cdot 7 - x \cdot 3 - 2 \cdot 21 - 3}{21} = 1$

Step 1.8

Simplify each term.

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Step 1.8.1

Multiply $7$ by $2$ .

$\frac{14 x - x \cdot 3 - 2 \cdot 21 - 3}{21} = 1$

Step 1.8.2

Multiply $3$ by $- 1$ .

$\frac{14 x - 3 x - 2 \cdot 21 - 3}{21} = 1$

Step 1.8.3

Multiply $- 2$ by $21$ .

$\frac{14 x - 3 x - 42 - 3}{21} = 1$

Step 1.9

Simplify by adding terms.

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Step 1.9.1

Subtract $3 x$ from $14 x$ .

$\frac{11 x - 42 - 3}{21} = 1$

Step 1.9.2

Subtract $3$ from $- 42$ .

$\frac{11 x - 45}{21} = 1$

Step 2

Multiply both sides by $21$ .

$\frac{11 x - 45}{21} \cdot 21 = 1 \cdot 21$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $21$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{11 x - 45}{21} \cdot 21 = 1 \cdot 21$

Step 3.1.1.2

Rewrite the expression.

$11 x - 45 = 1 \cdot 21$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $21$ by $1$ .

$11 x - 45 = 21$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $45$ to both sides of the equation.

$11 x = 21 + 45$

Step 4.1.2

Add $21$ and $45$ .

$11 x = 66$

Step 4.2

Divide each term in $11 x = 66$ by $11$ and simplify.

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Step 4.2.1

Divide each term in $11 x = 66$ by $11$ .

$\frac{11 x}{11} = \frac{66}{11}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $11$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{66}{11}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{66}{11}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $66$ by $11$ .

$x = 6$