Basic Math Examples

$\frac{a z - b}{b} = \frac{c - d}{d}$

Step 1

Multiply both sides by $b$ .

$\frac{a z - b}{b} b = \frac{c - d}{d} b$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $b$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{a z - b}{b} b = \frac{c - d}{d} b$

Step 2.1.1.2

Rewrite the expression.

$a z - b = \frac{c - d}{d} b$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{c - d}{d} b$ .

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Step 2.2.1.1

Combine $\frac{c - d}{d}$ and $b$ .

$a z - b = \frac{(c - d) b}{d}$

Step 2.2.1.2

Reorder factors in $\frac{(c - d) b}{d}$ .

$a z - b = \frac{b (c - d)}{d}$

Step 3

Solve for $a$ .

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Step 3.1

Add $b$ to both sides of the equation.

$a z = \frac{b (c - d)}{d} + b$

Step 3.2

Divide each term in $a z = \frac{b (c - d)}{d} + b$ by $z$ and simplify.

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Step 3.2.1

Divide each term in $a z = \frac{b (c - d)}{d} + b$ by $z$ .

$\frac{a z}{z} = \frac{\frac{b (c - d)}{d}}{z} + \frac{b}{z}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $z$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{a z}{z} = \frac{\frac{b (c - d)}{d}}{z} + \frac{b}{z}$

Step 3.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{\frac{b (c - d)}{d}}{z} + \frac{b}{z}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Combine the numerators over the common denominator.

$a = \frac{\frac{b (c - d)}{d} + b}{z}$

Step 3.2.3.2

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

$a = \frac{\frac{b (c - d)}{d} + \frac{b d}{d}}{z}$

Step 3.2.3.3

Combine the numerators over the common denominator.

$a = \frac{\frac{b (c - d) + b d}{d}}{z}$

Step 3.2.3.4

Simplify the numerator.

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Step 3.2.3.4.1

Factor $b$ out of $b (c - d) + b d$ .

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Step 3.2.3.4.1.1

Factor $b$ out of $b d$ .

$a = \frac{\frac{b (c - d) + b (d)}{d}}{z}$

Step 3.2.3.4.1.2

Factor $b$ out of $b (c - d) + b (d)$ .

$a = \frac{\frac{b (c - d + d)}{d}}{z}$

Step 3.2.3.4.2

Add $- d$ and $d$ .

$a = \frac{\frac{b (c + 0)}{d}}{z}$

Step 3.2.3.4.3

Add $c$ and $0$ .

$a = \frac{\frac{b c}{d}}{z}$

Step 3.2.3.5

Multiply the numerator by the reciprocal of the denominator.

$a = \frac{b c}{d} \cdot \frac{1}{z}$

Step 3.2.3.6

Multiply $\frac{b c}{d}$ by $\frac{1}{z}$ .

$a = \frac{b c}{d z}$