Precalculus Examples

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

Step 1

Simplify $\frac{2 (x + 2)}{3} - \frac{3 (3 x - 1)}{21}$ .

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Step 1.1

Cancel the common factors.

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Step 1.1.1

Factor $3$ out of $21$ .

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

Step 1.1.2

Cancel the common factor.

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

Step 1.1.3

Rewrite the expression.

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

Step 1.2

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

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

Step 1.3

To write $- \frac{3 x - 1}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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 $\frac{2 (x + 2)}{3}$ by $\frac{7}{7}$ .

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

Step 1.4.2

Multiply $3$ by $7$ .

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $7$ by $3$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Multiply $2$ by $2$ .

$\frac{(2 x + 4) \cdot 7 - (3 x - 1) \cdot 3}{21} = 1$

Step 1.6.3

Apply the distributive property.

$\frac{2 x \cdot 7 + 4 \cdot 7 - (3 x - 1) \cdot 3}{21} = 1$

Step 1.6.4

Multiply $7$ by $2$ .

$\frac{14 x + 4 \cdot 7 - (3 x - 1) \cdot 3}{21} = 1$

Step 1.6.5

Multiply $4$ by $7$ .

$\frac{14 x + 28 - (3 x - 1) \cdot 3}{21} = 1$

Step 1.6.6

Apply the distributive property.

$\frac{14 x + 28 + (- (3 x) - - 1) \cdot 3}{21} = 1$

Step 1.6.7

Multiply $3$ by $- 1$ .

$\frac{14 x + 28 + (- 3 x - - 1) \cdot 3}{21} = 1$

Step 1.6.8

Multiply $- 1$ by $- 1$ .

$\frac{14 x + 28 + (- 3 x + 1) \cdot 3}{21} = 1$

Step 1.6.9

Apply the distributive property.

$\frac{14 x + 28 - 3 x \cdot 3 + 1 \cdot 3}{21} = 1$

Step 1.6.10

Multiply $3$ by $- 3$ .

$\frac{14 x + 28 - 9 x + 1 \cdot 3}{21} = 1$

Step 1.6.11

Multiply $3$ by $1$ .

$\frac{14 x + 28 - 9 x + 3}{21} = 1$

Step 1.6.12

Subtract $9 x$ from $14 x$ .

$\frac{5 x + 28 + 3}{21} = 1$

Step 1.6.13

Add $28$ and $3$ .

$\frac{5 x + 31}{21} = 1$

Step 2

Multiply both sides by $21$ .

$\frac{5 x + 31}{21} \cdot 21 = 1 \cdot 21$

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.

$\frac{5 x + 31}{21} \cdot 21 = 1 \cdot 21$

Step 3.1.1.2

Rewrite the expression.

$5 x + 31 = 1 \cdot 21$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $21$ by $1$ .

$5 x + 31 = 21$

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

Subtract $31$ from both sides of the equation.

$5 x = 21 - 31$

Step 4.1.2

Subtract $31$ from $21$ .

$5 x = - 10$

Step 4.2

Divide each term in $5 x = - 10$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x = - 10$ by $5$ .

$\frac{5 x}{5} = \frac{- 10}{5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 10}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{5}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 10$ by $5$ .

$x = - 2$