Algebra Examples

$\frac{5}{6} (x - 4) = \frac{1}{2} (x - 2)$

Step 1

Simplify $\frac{5}{6} \cdot (x - 4)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{5}{6} \cdot (x - 4) = \frac{1}{2} \cdot (x - 2)$

Step 1.2

Simplify by adding zeros.

$\frac{5}{6} \cdot (x - 4) = \frac{1}{2} \cdot (x - 2)$

Step 1.3

Apply the distributive property.

$\frac{5}{6} x + \frac{5}{6} \cdot - 4 = \frac{1}{2} \cdot (x - 2)$

Step 1.4

Combine $\frac{5}{6}$ and $x$ .

$\frac{5 x}{6} + \frac{5}{6} \cdot - 4 = \frac{1}{2} \cdot (x - 2)$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $6$ .

$\frac{5 x}{6} + \frac{5}{2 (3)} \cdot - 4 = \frac{1}{2} \cdot (x - 2)$

Step 1.5.2

Factor $2$ out of $- 4$ .

$\frac{5 x}{6} + \frac{5}{2 \cdot 3} \cdot (2 \cdot - 2) = \frac{1}{2} \cdot (x - 2)$

Step 1.5.3

Cancel the common factor.

$\frac{5 x}{6} + \frac{5}{2 \cdot 3} \cdot (2 \cdot - 2) = \frac{1}{2} \cdot (x - 2)$

Step 1.5.4

Rewrite the expression.

$\frac{5 x}{6} + \frac{5}{3} \cdot - 2 = \frac{1}{2} \cdot (x - 2)$

Step 1.6

Combine $\frac{5}{3}$ and $- 2$ .

$\frac{5 x}{6} + \frac{5 \cdot - 2}{3} = \frac{1}{2} \cdot (x - 2)$

Step 1.7

Simplify the expression.

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Step 1.7.1

Multiply $5$ by $- 2$ .

$\frac{5 x}{6} + \frac{- 10}{3} = \frac{1}{2} \cdot (x - 2)$

Step 1.7.2

Move the negative in front of the fraction.

$\frac{5 x}{6} - \frac{10}{3} = \frac{1}{2} \cdot (x - 2)$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{5 x}{6} - \frac{10}{3} = \frac{1}{2} x + \frac{1}{2} \cdot - 2$

Step 2.2

Combine $\frac{1}{2}$ and $x$ .

$\frac{5 x}{6} - \frac{10}{3} = \frac{x}{2} + \frac{1}{2} \cdot - 2$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Factor $2$ out of $- 2$ .

$\frac{5 x}{6} - \frac{10}{3} = \frac{x}{2} + \frac{1}{2} \cdot (2 (- 1))$

Step 2.3.2

Cancel the common factor.

$\frac{5 x}{6} - \frac{10}{3} = \frac{x}{2} + \frac{1}{2} \cdot (2 \cdot - 1)$

Step 2.3.3

Rewrite the expression.

$\frac{5 x}{6} - \frac{10}{3} = \frac{x}{2} - 1$

Step 3

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

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Step 3.1

Subtract $\frac{x}{2}$ from both sides of the equation.

$\frac{5 x}{6} - \frac{10}{3} - \frac{x}{2} = - 1$

Step 3.2

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

$\frac{5 x}{6} - \frac{x}{2} \cdot \frac{3}{3} - \frac{10}{3} = - 1$

Step 3.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

$\frac{5 x}{6} - \frac{x \cdot 3}{2 \cdot 3} - \frac{10}{3} = - 1$

Step 3.3.2

Multiply $2$ by $3$ .

$\frac{5 x}{6} - \frac{x \cdot 3}{6} - \frac{10}{3} = - 1$

Step 3.4

Combine the numerators over the common denominator.

$\frac{5 x - x \cdot 3}{6} - \frac{10}{3} = - 1$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

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

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Step 3.5.1.1.1

Factor $x$ out of $5 x$ .

$\frac{x \cdot 5 - x \cdot 3}{6} - \frac{10}{3} = - 1$

Step 3.5.1.1.2

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

$\frac{x \cdot 5 + x (- 1 \cdot 3)}{6} - \frac{10}{3} = - 1$

Step 3.5.1.1.3

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

$\frac{x (5 - 1 \cdot 3)}{6} - \frac{10}{3} = - 1$

Step 3.5.1.2

Multiply $- 1$ by $3$ .

$\frac{x (5 - 3)}{6} - \frac{10}{3} = - 1$

Step 3.5.1.3

Subtract $3$ from $5$ .

$\frac{x \cdot 2}{6} - \frac{10}{3} = - 1$

Step 3.5.2

Cancel the common factor of $2$ and $6$ .

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Step 3.5.2.1

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

$\frac{2 \cdot x}{6} - \frac{10}{3} = - 1$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $2$ out of $6$ .

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

Step 3.5.2.2.2

Cancel the common factor.

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

Step 3.5.2.2.3

Rewrite the expression.

$\frac{x}{3} - \frac{10}{3} = - 1$

Step 4

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

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Step 4.1

Add $\frac{10}{3}$ to both sides of the equation.

$\frac{x}{3} = - 1 + \frac{10}{3}$

Step 4.2

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

$\frac{x}{3} = - 1 \cdot \frac{3}{3} + \frac{10}{3}$

Step 4.3

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{x}{3} = \frac{- 1 \cdot 3}{3} + \frac{10}{3}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{x}{3} = \frac{- 1 \cdot 3 + 10}{3}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $3$ .

$\frac{x}{3} = \frac{- 3 + 10}{3}$

Step 4.5.2

Add $- 3$ and $10$ .

$\frac{x}{3} = \frac{7}{3}$

Step 5

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

$x = 7$