Precalculus Examples

$\frac{x + 1}{3} + \frac{x + 2}{7} = 5$

Step 1

Simplify $\frac{x + 1}{3} + \frac{x + 2}{7}$ .

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Step 1.1

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

$\frac{x + 1}{3} \cdot \frac{7}{7} + \frac{x + 2}{7} = 5$

Step 1.2

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

$\frac{x + 1}{3} \cdot \frac{7}{7} + \frac{x + 2}{7} \cdot \frac{3}{3} = 5$

Step 1.3

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

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Step 1.3.1

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

$\frac{(x + 1) \cdot 7}{3 \cdot 7} + \frac{x + 2}{7} \cdot \frac{3}{3} = 5$

Step 1.3.2

Multiply $3$ by $7$ .

$\frac{(x + 1) \cdot 7}{21} + \frac{x + 2}{7} \cdot \frac{3}{3} = 5$

Step 1.3.3

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

$\frac{(x + 1) \cdot 7}{21} + \frac{(x + 2) \cdot 3}{7 \cdot 3} = 5$

Step 1.3.4

Multiply $7$ by $3$ .

$\frac{(x + 1) \cdot 7}{21} + \frac{(x + 2) \cdot 3}{21} = 5$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(x + 1) \cdot 7 + (x + 2) \cdot 3}{21} = 5$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{x \cdot 7 + 1 \cdot 7 + (x + 2) \cdot 3}{21} = 5$

Step 1.5.2

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

$\frac{7 \cdot x + 1 \cdot 7 + (x + 2) \cdot 3}{21} = 5$

Step 1.5.3

Multiply $7$ by $1$ .

$\frac{7 \cdot x + 7 + (x + 2) \cdot 3}{21} = 5$

Step 1.5.4

Apply the distributive property.

$\frac{7 x + 7 + x \cdot 3 + 2 \cdot 3}{21} = 5$

Step 1.5.5

Move $3$ to the left of $x$ .

$\frac{7 x + 7 + 3 \cdot x + 2 \cdot 3}{21} = 5$

Step 1.5.6

Multiply $2$ by $3$ .

$\frac{7 x + 7 + 3 \cdot x + 6}{21} = 5$

Step 1.5.7

Add $7 x$ and $3 x$ .

$\frac{10 x + 7 + 6}{21} = 5$

Step 1.5.8

Add $7$ and $6$ .

$\frac{10 x + 13}{21} = 5$

Step 2

Multiply both sides by $21$ .

$\frac{10 x + 13}{21} \cdot 21 = 5 \cdot 21$

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.

$\frac{10 x + 13}{21} \cdot 21 = 5 \cdot 21$

Step 3.1.1.2

Rewrite the expression.

$10 x + 13 = 5 \cdot 21$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $5$ by $21$ .

$10 x + 13 = 105$

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 $13$ from both sides of the equation.

$10 x = 105 - 13$

Step 4.1.2

Subtract $13$ from $105$ .

$10 x = 92$

Step 4.2

Divide each term in $10 x = 92$ by $10$ and simplify.

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Step 4.2.1

Divide each term in $10 x = 92$ by $10$ .

$\frac{10 x}{10} = \frac{92}{10}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $10$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{92}{10}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{92}{10}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $92$ and $10$ .

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Step 4.2.3.1.1

Factor $2$ out of $92$ .

$x = \frac{2 (46)}{10}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

Factor $2$ out of $10$ .

$x = \frac{2 \cdot 46}{2 \cdot 5}$

Step 4.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 46}{2 \cdot 5}$

Step 4.2.3.1.2.3

Rewrite the expression.

$x = \frac{46}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{46}{5}$

Decimal Form:

$x = 9.2$

Mixed Number Form:

$x = 9 \frac{1}{5}$