Algebra Examples

$S (x) = \frac{50 x}{100 - x}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$1, 100 - x$

Step 1.2

Remove parentheses.

$1, 100 - x$

Step 1.3

The LCM of one and any expression is the expression.

$100 - x$

Step 2

Multiply each term in $S x = \frac{50 x}{100 - x}$ by $100 - x$ to eliminate the fractions.

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Step 2.1

Multiply each term in $S x = \frac{50 x}{100 - x}$ by $100 - x$ .

$S x (100 - x) = \frac{50 x}{100 - x} (100 - x)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify by multiplying through.

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Step 2.2.1.1

Apply the distributive property.

$S x \cdot 100 + S x (- x) = \frac{50 x}{100 - x} (100 - x)$

Step 2.2.1.2

Reorder.

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Step 2.2.1.2.1

Move $100$ to the left of $S x$ .

$100 \cdot (S x) + S x (- x) = \frac{50 x}{100 - x} (100 - x)$

Step 2.2.1.2.2

Rewrite using the commutative property of multiplication.

$100 \cdot (S x) - S x \cdot x = \frac{50 x}{100 - x} (100 - x)$

Step 2.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.2.1

Move $x$ .

$100 S x - S (x \cdot x) = \frac{50 x}{100 - x} (100 - x)$

Step 2.2.2.2

Multiply $x$ by $x$ .

$100 S x - S x^{2} = \frac{50 x}{100 - x} (100 - x)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $100 - x$ .

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Step 2.3.1.1

Cancel the common factor.

$100 S x - S x^{2} = \frac{50 x}{100 - x} (100 - x)$

Step 2.3.1.2

Rewrite the expression.

$100 S x - S x^{2} = 50 x$

Step 3

Solve the equation.

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Step 3.1

Subtract $50 x$ from both sides of the equation.

$100 S x - S x^{2} - 50 x = 0$

Step 3.2

Factor $x$ out of $100 S x - S x^{2} - 50 x$ .

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Step 3.2.1

Factor $x$ out of $100 S x$ .

$x (100 S) - S x^{2} - 50 x = 0$

Step 3.2.2

Factor $x$ out of $- S x^{2}$ .

$x (100 S) + x (- S x) - 50 x = 0$

Step 3.2.3

Factor $x$ out of $- 50 x$ .

$x (100 S) + x (- S x) + x \cdot - 50 = 0$

Step 3.2.4

Factor $x$ out of $x (100 S) + x (- S x)$ .

$x (100 S - S x) + x \cdot - 50 = 0$

Step 3.2.5

Factor $x$ out of $x (100 S - S x) + x \cdot - 50$ .

$x (100 S - S x - 50) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$100 S - S x - 50 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $100 S - S x - 50$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $100 S - S x - 50$ equal to $0$ .

$100 S - S x - 50 = 0$

Step 3.5.2

Solve $100 S - S x - 50 = 0$ for $x$ .

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Step 3.5.2.1

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

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Step 3.5.2.1.1

Subtract $100 S$ from both sides of the equation.

$- S x - 50 = - 100 S$

Step 3.5.2.1.2

Add $50$ to both sides of the equation.

$- S x = - 100 S + 50$

Step 3.5.2.2

Divide each term in $- S x = - 100 S + 50$ by $- S$ and simplify.

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Step 3.5.2.2.1

Divide each term in $- S x = - 100 S + 50$ by $- S$ .

$\frac{- S x}{- S} = \frac{- 100 S}{- S} + \frac{50}{- S}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{S x}{S} = \frac{- 100 S}{- S} + \frac{50}{- S}$

Step 3.5.2.2.2.2

Cancel the common factor of $S$ .

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Step 3.5.2.2.2.2.1

Cancel the common factor.

$\frac{S x}{S} = \frac{- 100 S}{- S} + \frac{50}{- S}$

Step 3.5.2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 100 S}{- S} + \frac{50}{- S}$

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Simplify each term.

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Step 3.5.2.2.3.1.1

Cancel the common factor of $S$ .

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Step 3.5.2.2.3.1.1.1

Cancel the common factor.

$x = \frac{- 100 S}{- S} + \frac{50}{- S}$

Step 3.5.2.2.3.1.1.2

Rewrite the expression.

$x = \frac{- 100}{- 1} + \frac{50}{- S}$

Step 3.5.2.2.3.1.1.3

Move the negative one from the denominator of $\frac{- 100}{- 1}$ .

$x = - 1 \cdot - 100 + \frac{50}{- S}$

Step 3.5.2.2.3.1.2

Rewrite $- 1 \cdot - 100$ as $- - 100$ .

$x = - - 100 + \frac{50}{- S}$

Step 3.5.2.2.3.1.3

Multiply $- 1$ by $- 100$ .

$x = 100 + \frac{50}{- S}$

Step 3.5.2.2.3.1.4

Move the negative in front of the fraction.

$x = 100 - \frac{50}{S}$

Step 3.6

The final solution is all the values that make $x (100 S - S x - 50) = 0$ true.

$x = 0$

$x = 100 - \frac{50}{S}$

$x = 0$

$x = 100 - \frac{50}{S}$

Step 4

To rewrite as a function of $S$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $S$ is on the other side.

$f (S) = 0, 100 - \frac{50}{S}$