Basic Math Examples

$\frac{z}{3} + 5 = \frac{z}{7} + 1$

Step 1

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

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Step 1.1

Subtract $\frac{z}{7}$ from both sides of the equation.

$\frac{z}{3} + 5 - \frac{z}{7} = 1$

Step 1.2

To write $\frac{z}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{z}{3} \cdot \frac{7}{7} - \frac{z}{7} + 5 = 1$

Step 1.3

To write $- \frac{z}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{z}{3} \cdot \frac{7}{7} - \frac{z}{7} \cdot \frac{3}{3} + 5 = 1$

Step 1.4

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{z}{3}$ by $\frac{7}{7}$ .

$\frac{z \cdot 7}{3 \cdot 7} - \frac{z}{7} \cdot \frac{3}{3} + 5 = 1$

Step 1.4.2

Multiply $3$ by $7$ .

$\frac{z \cdot 7}{21} - \frac{z}{7} \cdot \frac{3}{3} + 5 = 1$

Step 1.4.3

Multiply $\frac{z}{7}$ by $\frac{3}{3}$ .

$\frac{z \cdot 7}{21} - \frac{z \cdot 3}{7 \cdot 3} + 5 = 1$

Step 1.4.4

Multiply $7$ by $3$ .

$\frac{z \cdot 7}{21} - \frac{z \cdot 3}{21} + 5 = 1$

Step 1.5

Combine the numerators over the common denominator.

$\frac{z \cdot 7 - z \cdot 3}{21} + 5 = 1$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $z$ out of $z \cdot 7 - z \cdot 3$ .

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Step 1.6.1.1.1

Factor $z$ out of $z \cdot 7$ .

$\frac{z (7) - z \cdot 3}{21} + 5 = 1$

Step 1.6.1.1.2

Factor $z$ out of $- z \cdot 3$ .

$\frac{z (7) + z (- 1 \cdot 3)}{21} + 5 = 1$

Step 1.6.1.1.3

Factor $z$ out of $z (7) + z (- 1 \cdot 3)$ .

$\frac{z (7 - 1 \cdot 3)}{21} + 5 = 1$

Step 1.6.1.2

Multiply $- 1$ by $3$ .

$\frac{z (7 - 3)}{21} + 5 = 1$

Step 1.6.1.3

Subtract $3$ from $7$ .

$\frac{z \cdot 4}{21} + 5 = 1$

Step 1.6.2

Move $4$ to the left of $z$ .

$\frac{4 z}{21} + 5 = 1$

Step 2

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

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Step 2.1

Subtract $5$ from both sides of the equation.

$\frac{4 z}{21} = 1 - 5$

Step 2.2

Subtract $5$ from $1$ .

$\frac{4 z}{21} = - 4$

Step 3

Multiply both sides of the equation by $\frac{21}{4}$ .

$\frac{21}{4} \cdot \frac{4 z}{21} = \frac{21}{4} \cdot - 4$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{21}{4} \cdot \frac{4 z}{21}$ .

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Step 4.1.1.1

Cancel the common factor of $21$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{21}{4} \cdot \frac{4 z}{21} = \frac{21}{4} \cdot - 4$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 z) = \frac{21}{4} \cdot - 4$

Step 4.1.1.2

Cancel the common factor of $4$ .

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Step 4.1.1.2.1

Factor $4$ out of $4 z$ .

$\frac{1}{4} (4 (z)) = \frac{21}{4} \cdot - 4$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 z) = \frac{21}{4} \cdot - 4$

Step 4.1.1.2.3

Rewrite the expression.

$z = \frac{21}{4} \cdot - 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{21}{4} \cdot - 4$ .

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Step 4.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1

Factor $4$ out of $- 4$ .

$z = \frac{21}{4} \cdot (4 (- 1))$

Step 4.2.1.1.2

Cancel the common factor.

$z = \frac{21}{4} \cdot (4 \cdot - 1)$

Step 4.2.1.1.3

Rewrite the expression.

$z = 21 \cdot - 1$

Step 4.2.1.2

Multiply $21$ by $- 1$ .

$z = - 21$