Calculus Examples

$x = \frac{60}{1 + \frac{5}{y + 1}}$

Step 1

Rewrite the equation as $\frac{60}{1 + \frac{5}{y + 1}} = x$ .

$\frac{60}{1 + \frac{5}{y + 1}} = x$

Step 2

Factor each term.

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Step 2.1

Write $1$ as a fraction with a common denominator.

$\frac{60}{\frac{y + 1}{y + 1} + \frac{5}{y + 1}} = x$

Step 2.2

Combine the numerators over the common denominator.

$\frac{60}{\frac{y + 1 + 5}{y + 1}} = x$

Step 2.3

Add $1$ and $5$ .

$\frac{60}{\frac{y + 6}{y + 1}} = x$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$60 \frac{y + 1}{y + 6} = x$

Step 2.5

Combine $60$ and $\frac{y + 1}{y + 6}$ .

$\frac{60 (y + 1)}{y + 6} = x$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y + 6, 1$

Step 3.2

Remove parentheses.

$y + 6, 1$

Step 3.3

The LCM of one and any expression is the expression.

$y + 6$

Step 4

Multiply each term in $\frac{60 (y + 1)}{y + 6} = x$ by $y + 6$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{60 (y + 1)}{y + 6} = x$ by $y + 6$ .

$\frac{60 (y + 1)}{y + 6} (y + 6) = x (y + 6)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $y + 6$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{60 (y + 1)}{y + 6} (y + 6) = x (y + 6)$

Step 4.2.1.2

Rewrite the expression.

$60 (y + 1) = x (y + 6)$

Step 4.2.2

Apply the distributive property.

$60 y + 60 \cdot 1 = x (y + 6)$

Step 4.2.3

Multiply $60$ by $1$ .

$60 y + 60 = x (y + 6)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$60 y + 60 = x y + x \cdot 6$

Step 4.3.2

Move $6$ to the left of $x$ .

$60 y + 60 = x y + 6 x$

Step 5

Solve the equation.

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Step 5.1

Subtract $x y$ from both sides of the equation.

$60 y + 60 - x y = 6 x$

Step 5.2

Subtract $60$ from both sides of the equation.

$60 y - x y = 6 x - 60$

Step 5.3

Factor $y$ out of $60 y - x y$ .

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Step 5.3.1

Factor $y$ out of $60 y$ .

$y \cdot 60 - x y = 6 x - 60$

Step 5.3.2

Factor $y$ out of $- x y$ .

$y \cdot 60 + y (- x) = 6 x - 60$

Step 5.3.3

Factor $y$ out of $y \cdot 60 + y (- x)$ .

$y (60 - x) = 6 x - 60$

Step 5.4

Divide each term in $y (60 - x) = 6 x - 60$ by $60 - x$ and simplify.

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Step 5.4.1

Divide each term in $y (60 - x) = 6 x - 60$ by $60 - x$ .

$\frac{y (60 - x)}{60 - x} = \frac{6 x}{60 - x} + \frac{- 60}{60 - x}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $60 - x$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y (60 - x)}{60 - x} = \frac{6 x}{60 - x} + \frac{- 60}{60 - x}$

Step 5.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{6 x}{60 - x} + \frac{- 60}{60 - x}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y = \frac{6 x - 60}{60 - x}$

Step 5.4.3.2

Factor $6$ out of $6 x - 60$ .

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Step 5.4.3.2.1

Factor $6$ out of $6 x$ .

$y = \frac{6 (x) - 60}{60 - x}$

Step 5.4.3.2.2

Factor $6$ out of $- 60$ .

$y = \frac{6 x + 6 \cdot - 10}{60 - x}$

Step 5.4.3.2.3

Factor $6$ out of $6 x + 6 \cdot - 10$ .

$y = \frac{6 (x - 10)}{60 - x}$