Algebra Examples

$\frac{x}{2} + \frac{x}{3} = \frac{5 x + 2}{6}$

Step 1

Multiply both sides by $6$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x}{2} + \frac{x}{3}) \cdot 6$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

Multiply $2$ by $3$ .

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

Step 2.1.1.3.3

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

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

Step 2.1.1.3.4

Multiply $3$ by $2$ .

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

Step 2.1.1.4

Combine the numerators over the common denominator.

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

Step 2.1.1.5

Simplify the numerator.

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Step 2.1.1.5.1

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

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

Step 2.1.1.5.2

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

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

Step 2.1.1.5.3

Add $3 x$ and $2 x$ .

$\frac{5 x}{6} \cdot 6 = \frac{5 x + 2}{6} \cdot 6$

Step 2.1.1.6

Cancel the common factor of $6$ .

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Step 2.1.1.6.1

Cancel the common factor.

$\frac{5 x}{6} \cdot 6 = \frac{5 x + 2}{6} \cdot 6$

Step 2.1.1.6.2

Rewrite the expression.

$5 x = \frac{5 x + 2}{6} \cdot 6$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$5 x = \frac{5 x + 2}{6} \cdot 6$

Step 2.2.1.2

Rewrite the expression.

$5 x = 5 x + 2$

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 $5 x$ from both sides of the equation.

$5 x - 5 x = 2$

Step 3.1.2

Subtract $5 x$ from $5 x$ .

$0 = 2$

Step 3.2

Since $0 \neq 2$ , there are no solutions.

No solution