Basic Math Examples

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

Step 1

Simplify $\frac{1}{2} z + 2 (\frac{3}{4} z - 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

Apply the distributive property.

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

Step 1.1.4

Cancel the common factor of $2$ .

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Step 1.1.4.1

Factor $2$ out of $4$ .

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

Step 1.1.4.2

Cancel the common factor.

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

Step 1.1.4.3

Rewrite the expression.

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

Step 1.1.5

Multiply $2$ by $- 1$ .

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

Step 1.2

Simplify terms.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Add $z$ and $3 z$ .

$- 2 + \frac{4 z}{2} = \frac{1}{4} z + 6$

Step 1.2.3

Cancel the common factor of $4$ and $2$ .

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Step 1.2.3.1

Factor $2$ out of $4 z$ .

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

Step 1.2.3.2

Cancel the common factors.

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Step 1.2.3.2.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.2

Cancel the common factor.

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

Step 1.2.3.2.3

Rewrite the expression.

$- 2 + \frac{2 z}{1} = \frac{1}{4} z + 6$

Step 1.2.3.2.4

Divide $2 z$ by $1$ .

$- 2 + 2 z = \frac{1}{4} z + 6$

Step 2

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

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

Step 3

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

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Step 3.1

Subtract $\frac{z}{4}$ from both sides of the equation.

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

Step 3.2

To write $2 z$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3.3

Combine $2 z$ and $\frac{4}{4}$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

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 $z$ out of $2 z \cdot 4 - z$ .

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Step 3.5.1.1.1

Factor $z$ out of $2 z \cdot 4$ .

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

Step 3.5.1.1.2

Factor $z$ out of $- z$ .

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

Step 3.5.1.1.3

Factor $z$ out of $z (2 \cdot 4) + z \cdot - 1$ .

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

Step 3.5.1.2

Multiply $2$ by $4$ .

$- 2 + \frac{z (8 - 1)}{4} = 6$

Step 3.5.1.3

Subtract $1$ from $8$ .

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

Step 3.5.2

Move $7$ to the left of $z$ .

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

Step 4

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

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Step 4.1

Add $2$ to both sides of the equation.

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

Step 4.2

Add $6$ and $2$ .

$\frac{7 z}{4} = 8$

Step 5

Multiply both sides of the equation by $\frac{4}{7}$ .

$\frac{4}{7} \cdot \frac{7 z}{4} = \frac{4}{7} \cdot 8$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{4}{7} \cdot \frac{7 z}{4}$ .

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Step 6.1.1.1

Cancel the common factor of $4$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{4}{7} \cdot \frac{7 z}{4} = \frac{4}{7} \cdot 8$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 z) = \frac{4}{7} \cdot 8$

Step 6.1.1.2

Cancel the common factor of $7$ .

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Step 6.1.1.2.1

Factor $7$ out of $7 z$ .

$\frac{1}{7} (7 (z)) = \frac{4}{7} \cdot 8$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 z) = \frac{4}{7} \cdot 8$

Step 6.1.1.2.3

Rewrite the expression.

$z = \frac{4}{7} \cdot 8$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $\frac{4}{7} \cdot 8$ .

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Step 6.2.1.1

Combine $\frac{4}{7}$ and $8$ .

$z = \frac{4 \cdot 8}{7}$

Step 6.2.1.2

Multiply $4$ by $8$ .

$z = \frac{32}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = \frac{32}{7}$

Decimal Form:

$z = 4.\overline{571428}$

Mixed Number Form:

$z = 4 \frac{4}{7}$