Algebra Examples

$\frac{x + 3}{10} + \frac{x + 4}{7} = \frac{2 x + 1}{70}$

Step 1

Multiply both sides by $70$ .

$(\frac{x + 3}{10} + \frac{x + 4}{7}) \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x + 3}{10} + \frac{x + 4}{7}) \cdot 70$ .

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Step 2.1.1.1

To write $\frac{x + 3}{10}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$(\frac{x + 3}{10} \cdot \frac{7}{7} + \frac{x + 4}{7}) \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.2

To write $\frac{x + 4}{7}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$(\frac{x + 3}{10} \cdot \frac{7}{7} + \frac{x + 4}{7} \cdot \frac{10}{10}) \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.3

Write each expression with a common denominator of $70$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.1.3.1

Multiply $\frac{x + 3}{10}$ by $\frac{7}{7}$ .

$(\frac{(x + 3) \cdot 7}{10 \cdot 7} + \frac{x + 4}{7} \cdot \frac{10}{10}) \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.3.2

Multiply $10$ by $7$ .

$(\frac{(x + 3) \cdot 7}{70} + \frac{x + 4}{7} \cdot \frac{10}{10}) \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.3.3

Multiply $\frac{x + 4}{7}$ by $\frac{10}{10}$ .

$(\frac{(x + 3) \cdot 7}{70} + \frac{(x + 4) \cdot 10}{7 \cdot 10}) \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.3.4

Multiply $7$ by $10$ .

$(\frac{(x + 3) \cdot 7}{70} + \frac{(x + 4) \cdot 10}{70}) \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.4

Combine the numerators over the common denominator.

$\frac{(x + 3) \cdot 7 + (x + 4) \cdot 10}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.5

Simplify the numerator.

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Step 2.1.1.5.1

Apply the distributive property.

$\frac{x \cdot 7 + 3 \cdot 7 + (x + 4) \cdot 10}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.5.2

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

$\frac{7 \cdot x + 3 \cdot 7 + (x + 4) \cdot 10}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.5.3

Multiply $3$ by $7$ .

$\frac{7 \cdot x + 21 + (x + 4) \cdot 10}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.5.4

Apply the distributive property.

$\frac{7 x + 21 + x \cdot 10 + 4 \cdot 10}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.5.5

Move $10$ to the left of $x$ .

$\frac{7 x + 21 + 10 \cdot x + 4 \cdot 10}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.5.6

Multiply $4$ by $10$ .

$\frac{7 x + 21 + 10 \cdot x + 40}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.5.7

Add $7 x$ and $10 x$ .

$\frac{17 x + 21 + 40}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.5.8

Add $21$ and $40$ .

$\frac{17 x + 61}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.6

Cancel the common factor of $70$ .

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Step 2.1.1.6.1

Cancel the common factor.

$\frac{17 x + 61}{70} \cdot 70 = \frac{2 x + 1}{70} \cdot 70$

Step 2.1.1.6.2

Rewrite the expression.

$17 x + 61 = \frac{2 x + 1}{70} \cdot 70$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $70$ .

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Step 2.2.1.1

Cancel the common factor.

$17 x + 61 = \frac{2 x + 1}{70} \cdot 70$

Step 2.2.1.2

Rewrite the expression.

$17 x + 61 = 2 x + 1$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $2 x$ from both sides of the equation.

$17 x + 61 - 2 x = 1$

Step 3.1.2

Subtract $2 x$ from $17 x$ .

$15 x + 61 = 1$

Step 3.2

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

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Step 3.2.1

Subtract $61$ from both sides of the equation.

$15 x = 1 - 61$

Step 3.2.2

Subtract $61$ from $1$ .

$15 x = - 60$

Step 3.3

Divide each term in $15 x = - 60$ by $15$ and simplify.

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Step 3.3.1

Divide each term in $15 x = - 60$ by $15$ .

$\frac{15 x}{15} = \frac{- 60}{15}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $15$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{15 x}{15} = \frac{- 60}{15}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 60}{15}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 60$ by $15$ .

$x = - 4$