Pre-Algebra Examples

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

Step 1

Simplify $\frac{2 x}{3} + x$ .

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Step 1.1

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

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

Step 1.2

Simplify terms.

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Step 1.2.1

Combine $x$ and $\frac{3}{3}$ .

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.3

Simplify the numerator.

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Step 1.3.1

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

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

Step 1.3.2

Add $2 x$ and $3 x$ .

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

Step 2

Simplify $10 - \frac{x - 3}{2}$ .

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Step 2.1

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $10$ and $\frac{2}{2}$ .

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

Step 2.2.2

Combine the numerators over the common denominator.

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

Multiply $10$ by $2$ .

$\frac{5 x}{3} = \frac{20 - (x - 3)}{2}$

Step 2.3.2

Apply the distributive property.

$\frac{5 x}{3} = \frac{20 - x - - 3}{2}$

Step 2.3.3

Multiply $- 1$ by $- 3$ .

$\frac{5 x}{3} = \frac{20 - x + 3}{2}$

Step 2.3.4

Add $20$ and $3$ .

$\frac{5 x}{3} = \frac{- x + 23}{2}$

Step 2.4

Simplify with factoring out.

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Step 2.4.1

Factor $- 1$ out of $- x$ .

$\frac{5 x}{3} = \frac{- (x) + 23}{2}$

Step 2.4.2

Rewrite $23$ as $- 1 (- 23)$ .

$\frac{5 x}{3} = \frac{- (x) - 1 (- 23)}{2}$

Step 2.4.3

Factor $- 1$ out of $- (x) - 1 (- 23)$ .

$\frac{5 x}{3} = \frac{- (x - 23)}{2}$

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Rewrite $- (x - 23)$ as $- 1 (x - 23)$ .

$\frac{5 x}{3} = \frac{- 1 (x - 23)}{2}$

Step 2.4.4.2

Move the negative in front of the fraction.

$\frac{5 x}{3} = - \frac{x - 23}{2}$

Step 3

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

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Step 3.1

Add $\frac{x - 23}{2}$ to both sides of the equation.

$\frac{5 x}{3} + \frac{x - 23}{2} = 0$

Step 3.2

To write $\frac{5 x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 x}{3} \cdot \frac{2}{2} + \frac{x - 23}{2} = 0$

Step 3.3

To write $\frac{x - 23}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 x}{3} \cdot \frac{2}{2} + \frac{x - 23}{2} \cdot \frac{3}{3} = 0$

Step 3.4

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

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Step 3.4.1

Multiply $\frac{5 x}{3}$ by $\frac{2}{2}$ .

$\frac{5 x \cdot 2}{3 \cdot 2} + \frac{x - 23}{2} \cdot \frac{3}{3} = 0$

Step 3.4.2

Multiply $3$ by $2$ .

$\frac{5 x \cdot 2}{6} + \frac{x - 23}{2} \cdot \frac{3}{3} = 0$

Step 3.4.3

Multiply $\frac{x - 23}{2}$ by $\frac{3}{3}$ .

$\frac{5 x \cdot 2}{6} + \frac{(x - 23) \cdot 3}{2 \cdot 3} = 0$

Step 3.4.4

Multiply $2$ by $3$ .

$\frac{5 x \cdot 2}{6} + \frac{(x - 23) \cdot 3}{6} = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{5 x \cdot 2 + (x - 23) \cdot 3}{6} = 0$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $5$ .

$\frac{10 x + (x - 23) \cdot 3}{6} = 0$

Step 3.6.2

Apply the distributive property.

$\frac{10 x + x \cdot 3 - 23 \cdot 3}{6} = 0$

Step 3.6.3

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

$\frac{10 x + 3 \cdot x - 23 \cdot 3}{6} = 0$

Step 3.6.4

Multiply $- 23$ by $3$ .

$\frac{10 x + 3 \cdot x - 69}{6} = 0$

Step 3.6.5

Add $10 x$ and $3 x$ .

$\frac{13 x - 69}{6} = 0$

Step 4

Set the numerator equal to zero.

$13 x - 69 = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Add $69$ to both sides of the equation.

$13 x = 69$

Step 5.2

Divide each term in $13 x = 69$ by $13$ and simplify.

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Step 5.2.1

Divide each term in $13 x = 69$ by $13$ .

$\frac{13 x}{13} = \frac{69}{13}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $13$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{13 x}{13} = \frac{69}{13}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{69}{13}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{69}{13}$

Decimal Form:

$x = 5.\overline{307692}$

Mixed Number Form:

$x = 5 \frac{4}{13}$