Basic Math Examples

$\frac{2 z + 3}{3} + \frac{3 z - 4}{6} = \frac{z - 2}{2}$

Step 1

Multiply both sides by $2$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{2 z + 3}{3} + \frac{3 z - 4}{6}) \cdot 2$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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Step 2.1.1.2.1

Multiply $\frac{2 z + 3}{3}$ by $\frac{2}{2}$ .

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

Step 2.1.1.2.2

Multiply $3$ by $2$ .

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

Step 2.1.1.3

Combine the numerators over the common denominator.

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

Step 2.1.1.4

Simplify the numerator.

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Step 2.1.1.4.1

Apply the distributive property.

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

Step 2.1.1.4.2

Multiply $2$ by $2$ .

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

Step 2.1.1.4.3

Multiply $3$ by $2$ .

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

Step 2.1.1.4.4

Add $4 z$ and $3 z$ .

$\frac{7 z + 6 - 4}{6} \cdot 2 = \frac{z - 2}{2} \cdot 2$

Step 2.1.1.4.5

Subtract $4$ from $6$ .

$\frac{7 z + 2}{6} \cdot 2 = \frac{z - 2}{2} \cdot 2$

Step 2.1.1.5

Cancel the common factor of $2$ .

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Step 2.1.1.5.1

Factor $2$ out of $6$ .

$\frac{7 z + 2}{2 (3)} \cdot 2 = \frac{z - 2}{2} \cdot 2$

Step 2.1.1.5.2

Cancel the common factor.

$\frac{7 z + 2}{2 \cdot 3} \cdot 2 = \frac{z - 2}{2} \cdot 2$

Step 2.1.1.5.3

Rewrite the expression.

$\frac{7 z + 2}{3} = \frac{z - 2}{2} \cdot 2$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{7 z + 2}{3} = \frac{z - 2}{2} \cdot 2$

Step 2.2.1.2

Rewrite the expression.

$\frac{7 z + 2}{3} = z - 2$

Step 3

Solve for $z$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{7 z + 2}{3} \cdot 3 = (z - 2) \cdot 3$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{7 z + 2}{3} \cdot 3 = (z - 2) \cdot 3$

Step 3.2.1.1.2

Rewrite the expression.

$7 z + 2 = (z - 2) \cdot 3$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(z - 2) \cdot 3$ .

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Step 3.2.2.1.1

Apply the distributive property.

$7 z + 2 = z \cdot 3 - 2 \cdot 3$

Step 3.2.2.1.2

Simplify the expression.

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Step 3.2.2.1.2.1

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

$7 z + 2 = 3 \cdot z - 2 \cdot 3$

Step 3.2.2.1.2.2

Multiply $- 2$ by $3$ .

$7 z + 2 = 3 z - 6$

Step 3.3

Solve for $z$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $3 z$ from both sides of the equation.

$7 z + 2 - 3 z = - 6$

Step 3.3.1.2

Subtract $3 z$ from $7 z$ .

$4 z + 2 = - 6$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $2$ from both sides of the equation.

$4 z = - 6 - 2$

Step 3.3.2.2

Subtract $2$ from $- 6$ .

$4 z = - 8$

Step 3.3.3

Divide each term in $4 z = - 8$ by $4$ and simplify.

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Step 3.3.3.1

Divide each term in $4 z = - 8$ by $4$ .

$\frac{4 z}{4} = \frac{- 8}{4}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{4 z}{4} = \frac{- 8}{4}$

Step 3.3.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 8}{4}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $- 8$ by $4$ .

$z = - 2$