Algebra Examples

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{2}{7} (x - \frac{1}{14})$

Step 1

Simplify $1 - \frac{2}{7} \cdot (x - \frac{1}{14})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{2}{7} x - \frac{2}{7} (- \frac{1}{14})$

Step 1.1.2

Combine $x$ and $\frac{2}{7}$ .

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} - \frac{2}{7} (- \frac{1}{14})$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Move the leading negative in $- \frac{2}{7}$ into the numerator.

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{- 2}{7} (- \frac{1}{14})$

Step 1.1.3.2

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{- 2}{7} \cdot \frac{- 1}{14}$

Step 1.1.3.3

Factor $2$ out of $- 2$ .

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{2 (- 1)}{7} \cdot \frac{- 1}{14}$

Step 1.1.3.4

Factor $2$ out of $14$ .

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{2 \cdot - 1}{7} \cdot \frac{- 1}{2 \cdot 7}$

Step 1.1.3.5

Cancel the common factor.

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{2 \cdot - 1}{7} \cdot \frac{- 1}{2 \cdot 7}$

Step 1.1.3.6

Rewrite the expression.

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{- 1}{7} \cdot \frac{- 1}{7}$

Step 1.1.4

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

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{- - 1}{7 \cdot 7}$

Step 1.1.5

Multiply $- 1$ by $- 1$ .

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{1}{7 \cdot 7}$

Step 1.1.6

Multiply $7$ by $7$ .

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{x \cdot 2}{7} + \frac{1}{49}$

Step 1.1.7

Move $2$ to the left of $x$ .

$\frac{7 x}{4} - \frac{1}{7} = 1 - \frac{2 x}{7} + \frac{1}{49}$

Step 1.2

Simplify the expression.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$\frac{7 x}{4} - \frac{1}{7} = - \frac{2 x}{7} + \frac{49}{49} + \frac{1}{49}$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{7 x}{4} - \frac{1}{7} = - \frac{2 x}{7} + \frac{49 + 1}{49}$

Step 1.2.3

Add $49$ and $1$ .

$\frac{7 x}{4} - \frac{1}{7} = - \frac{2 x}{7} + \frac{50}{49}$

Step 2

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

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Step 2.1

Add $\frac{2 x}{7}$ to both sides of the equation.

$\frac{7 x}{4} - \frac{1}{7} + \frac{2 x}{7} = \frac{50}{49}$

Step 2.2

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

$\frac{7 x}{4} \cdot \frac{7}{7} + \frac{2 x}{7} - \frac{1}{7} = \frac{50}{49}$

Step 2.3

To write $\frac{2 x}{7}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{7 x}{4} \cdot \frac{7}{7} + \frac{2 x}{7} \cdot \frac{4}{4} - \frac{1}{7} = \frac{50}{49}$

Step 2.4

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

$\frac{7 x \cdot 7}{4 \cdot 7} + \frac{2 x}{7} \cdot \frac{4}{4} - \frac{1}{7} = \frac{50}{49}$

Step 2.4.2

Multiply $4$ by $7$ .

$\frac{7 x \cdot 7}{28} + \frac{2 x}{7} \cdot \frac{4}{4} - \frac{1}{7} = \frac{50}{49}$

Step 2.4.3

Multiply $\frac{2 x}{7}$ by $\frac{4}{4}$ .

$\frac{7 x \cdot 7}{28} + \frac{2 x \cdot 4}{7 \cdot 4} - \frac{1}{7} = \frac{50}{49}$

Step 2.4.4

Multiply $7$ by $4$ .

$\frac{7 x \cdot 7}{28} + \frac{2 x \cdot 4}{28} - \frac{1}{7} = \frac{50}{49}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{7 x \cdot 7 + 2 x \cdot 4}{28} - \frac{1}{7} = \frac{50}{49}$

Step 2.6

Simplify each term.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Factor $x$ out of $7 x \cdot 7 + 2 x \cdot 4$ .

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Step 2.6.1.1.1

Factor $x$ out of $7 x \cdot 7$ .

$\frac{x (7 \cdot 7) + 2 x \cdot 4}{28} - \frac{1}{7} = \frac{50}{49}$

Step 2.6.1.1.2

Factor $x$ out of $2 x \cdot 4$ .

$\frac{x (7 \cdot 7) + x (2 \cdot 4)}{28} - \frac{1}{7} = \frac{50}{49}$

Step 2.6.1.1.3

Factor $x$ out of $x (7 \cdot 7) + x (2 \cdot 4)$ .

$\frac{x (7 \cdot 7 + 2 \cdot 4)}{28} - \frac{1}{7} = \frac{50}{49}$

Step 2.6.1.2

Multiply $7$ by $7$ .

$\frac{x (49 + 2 \cdot 4)}{28} - \frac{1}{7} = \frac{50}{49}$

Step 2.6.1.3

Multiply $2$ by $4$ .

$\frac{x (49 + 8)}{28} - \frac{1}{7} = \frac{50}{49}$

Step 2.6.1.4

Add $49$ and $8$ .

$\frac{x \cdot 57}{28} - \frac{1}{7} = \frac{50}{49}$

Step 2.6.2

Move $57$ to the left of $x$ .

$\frac{57 x}{28} - \frac{1}{7} = \frac{50}{49}$

Step 3

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

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Step 3.1

Add $\frac{1}{7}$ to both sides of the equation.

$\frac{57 x}{28} = \frac{50}{49} + \frac{1}{7}$

Step 3.2

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

$\frac{57 x}{28} = \frac{50}{49} + \frac{1}{7} \cdot \frac{7}{7}$

Step 3.3

Write each expression with a common denominator of $49$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

$\frac{57 x}{28} = \frac{50}{49} + \frac{7}{7 \cdot 7}$

Step 3.3.2

Multiply $7$ by $7$ .

$\frac{57 x}{28} = \frac{50}{49} + \frac{7}{49}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{57 x}{28} = \frac{50 + 7}{49}$

Step 3.5

Add $50$ and $7$ .

$\frac{57 x}{28} = \frac{57}{49}$

Step 4

Multiply both sides of the equation by $\frac{28}{57}$ .

$\frac{28}{57} \cdot \frac{57 x}{28} = \frac{28}{57} \cdot \frac{57}{49}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{28}{57} \cdot \frac{57 x}{28}$ .

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Step 5.1.1.1

Cancel the common factor of $28$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{28}{57} \cdot \frac{57 x}{28} = \frac{28}{57} \cdot \frac{57}{49}$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{57} (57 x) = \frac{28}{57} \cdot \frac{57}{49}$

Step 5.1.1.2

Cancel the common factor of $57$ .

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Step 5.1.1.2.1

Factor $57$ out of $57 x$ .

$\frac{1}{57} (57 (x)) = \frac{28}{57} \cdot \frac{57}{49}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{57} (57 x) = \frac{28}{57} \cdot \frac{57}{49}$

Step 5.1.1.2.3

Rewrite the expression.

$x = \frac{28}{57} \cdot \frac{57}{49}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{28}{57} \cdot \frac{57}{49}$ .

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Step 5.2.1.1

Cancel the common factor of $7$ .

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Step 5.2.1.1.1

Factor $7$ out of $28$ .

$x = \frac{7 (4)}{57} \cdot \frac{57}{49}$

Step 5.2.1.1.2

Factor $7$ out of $49$ .

$x = \frac{7 \cdot 4}{57} \cdot \frac{57}{7 \cdot 7}$

Step 5.2.1.1.3

Cancel the common factor.

$x = \frac{7 \cdot 4}{57} \cdot \frac{57}{7 \cdot 7}$

Step 5.2.1.1.4

Rewrite the expression.

$x = \frac{4}{57} \cdot \frac{57}{7}$

Step 5.2.1.2

Cancel the common factor of $57$ .

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Step 5.2.1.2.1

Cancel the common factor.

$x = \frac{4}{57} \cdot \frac{57}{7}$

Step 5.2.1.2.2

Rewrite the expression.

$x = 4 (\frac{1}{7})$

Step 5.2.1.3

Combine $4$ and $\frac{1}{7}$ .

$x = \frac{4}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{7}$

Decimal Form:

$x = 0.\overline{571428}$