Algebra Examples

$\frac{3 x}{5} - x = \frac{x}{10} - \frac{5}{2}$

Step 1

Simplify $\frac{3 x}{5} - x$ .

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Step 1.1

To write $- x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3 x}{5} - x \cdot \frac{5}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.2

Simplify terms.

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Step 1.2.1

Combine $- x$ and $\frac{5}{5}$ .

$\frac{3 x}{5} + \frac{- x \cdot 5}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{3 x - x \cdot 5}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

Factor $x$ out of $3 x - x \cdot 5$ .

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Step 1.3.1.1

Factor $x$ out of $3 x$ .

$\frac{x \cdot 3 - x \cdot 5}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.3.1.2

Factor $x$ out of $- x \cdot 5$ .

$\frac{x \cdot 3 + x (- 1 \cdot 5)}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.3.1.3

Factor $x$ out of $x \cdot 3 + x (- 1 \cdot 5)$ .

$\frac{x (3 - 1 \cdot 5)}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.3.2

Multiply $- 1$ by $5$ .

$\frac{x (3 - 5)}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.3.3

Subtract $5$ from $3$ .

$\frac{x \cdot - 2}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.4

Simplify the expression.

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Step 1.4.1

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

$\frac{- 2 \cdot x}{5} = \frac{x}{10} - \frac{5}{2}$

Step 1.4.2

Move the negative in front of the fraction.

$- \frac{2 x}{5} = \frac{x}{10} - \frac{5}{2}$

Step 2

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

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Step 2.1

Subtract $\frac{x}{10}$ from both sides of the equation.

$- \frac{2 x}{5} - \frac{x}{10} = - \frac{5}{2}$

Step 2.2

To write $- \frac{2 x}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{2 x}{5} \cdot \frac{2}{2} - \frac{x}{10} = - \frac{5}{2}$

Step 2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{2 x}{5}$ by $\frac{2}{2}$ .

$- \frac{2 x \cdot 2}{5 \cdot 2} - \frac{x}{10} = - \frac{5}{2}$

Step 2.3.2

Multiply $5$ by $2$ .

$- \frac{2 x \cdot 2}{10} - \frac{x}{10} = - \frac{5}{2}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{- 2 x \cdot 2 - x}{10} = - \frac{5}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

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

$\frac{x (- 2 \cdot 2) - x}{10} = - \frac{5}{2}$

Step 2.5.1.2

Factor $x$ out of $- x$ .

$\frac{x (- 2 \cdot 2) + x \cdot - 1}{10} = - \frac{5}{2}$

Step 2.5.1.3

Factor $x$ out of $x (- 2 \cdot 2) + x \cdot - 1$ .

$\frac{x (- 2 \cdot 2 - 1)}{10} = - \frac{5}{2}$

Step 2.5.2

Multiply $- 2$ by $2$ .

$\frac{x (- 4 - 1)}{10} = - \frac{5}{2}$

Step 2.5.3

Subtract $1$ from $- 4$ .

$\frac{x \cdot - 5}{10} = - \frac{5}{2}$

Step 2.6

Cancel the common factor of $- 5$ and $10$ .

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Step 2.6.1

Factor $5$ out of $x \cdot - 5$ .

$\frac{5 (x \cdot - 1)}{10} = - \frac{5}{2}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $5$ out of $10$ .

$\frac{5 (x \cdot - 1)}{5 (2)} = - \frac{5}{2}$

Step 2.6.2.2

Cancel the common factor.

$\frac{5 (x \cdot - 1)}{5 \cdot 2} = - \frac{5}{2}$

Step 2.6.2.3

Rewrite the expression.

$\frac{x \cdot - 1}{2} = - \frac{5}{2}$

Step 2.7

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{2} = - \frac{5}{2}$

Step 2.8

Move the negative in front of the fraction.

$- \frac{x}{2} = - \frac{5}{2}$

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x = - 5$

Step 4

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x = - 5$ by $- 1$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

Divide $x$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 5$ by $- 1$ .

$x = 5$