Algebra Examples

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

Step 1

Simplify $\frac{1}{3} \cdot (z + 4) - 6$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

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

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

Step 1.1.3

Combine $\frac{1}{3}$ and $4$ .

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

Step 1.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.3

Combine $- 6$ and $\frac{3}{3}$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 6$ by $3$ .

$\frac{z}{3} + \frac{4 - 18}{3} = \frac{2}{3} \cdot (5 - z)$

Step 1.5.2

Subtract $18$ from $4$ .

$\frac{z}{3} + \frac{- 14}{3} = \frac{2}{3} \cdot (5 - z)$

Step 1.6

Move the negative in front of the fraction.

$\frac{z}{3} - \frac{14}{3} = \frac{2}{3} \cdot (5 - z)$

Step 2

Simplify $\frac{2}{3} \cdot (5 - z)$ .

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Step 2.1

Apply the distributive property.

$\frac{z}{3} - \frac{14}{3} = \frac{2}{3} \cdot 5 + \frac{2}{3} (- z)$

Step 2.2

Multiply $\frac{2}{3} \cdot 5$ .

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Step 2.2.1

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

$\frac{z}{3} - \frac{14}{3} = \frac{2 \cdot 5}{3} + \frac{2}{3} (- z)$

Step 2.2.2

Multiply $2$ by $5$ .

$\frac{z}{3} - \frac{14}{3} = \frac{10}{3} + \frac{2}{3} (- z)$

Step 2.3

Combine $\frac{2}{3}$ and $z$ .

$\frac{z}{3} - \frac{14}{3} = \frac{10}{3} - \frac{2 z}{3}$

Step 3

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

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Step 3.1

Add $\frac{2 z}{3}$ to both sides of the equation.

$\frac{z}{3} - \frac{14}{3} + \frac{2 z}{3} = \frac{10}{3}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{z - 14 + 2 z}{3} = \frac{10}{3}$

Step 3.3

Add $z$ and $2 z$ .

$\frac{3 z - 14}{3} = \frac{10}{3}$

Step 4

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

$3 z - 14 = 10$

Step 5

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

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Step 5.1

Add $14$ to both sides of the equation.

$3 z = 10 + 14$

Step 5.2

Add $10$ and $14$ .

$3 z = 24$

Step 6

Divide each term in $3 z = 24$ by $3$ and simplify.

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Step 6.1

Divide each term in $3 z = 24$ by $3$ .

$\frac{3 z}{3} = \frac{24}{3}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{24}{3}$

Step 6.2.1.2

Divide $z$ by $1$ .

$z = \frac{24}{3}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $24$ by $3$ .

$z = 8$