Algebra Examples

$y = \frac{100 (0.02) + x (0.005)}{100 + x}$

Step 1

Rewrite the equation as $\frac{100 (0.02) + x (0.005)}{100 + x} = y$ .

$\frac{100 (0.02) + x (0.005)}{100 + x} = y$

Step 2

Factor each term.

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Step 2.1

Factor $0.000025$ out of $100 (0.02) + x (0.005)$ .

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Step 2.1.1

Reorder $x$ and $0.005$ .

$\frac{100 (0.02) + 0.005 x}{100 + x} = y$

Step 2.1.2

Factor $0.000025$ out of $100 (0.02)$ .

$\frac{0.000025 (4000000 \cdot 0.02) + 0.005 x}{100 + x} = y$

Step 2.1.3

Factor $0.000025$ out of $0.005 x$ .

$\frac{0.000025 (4000000 \cdot 0.02) + 0.000025 (200 x)}{100 + x} = y$

Step 2.1.4

Factor $0.000025$ out of $0.000025 (4000000 \cdot 0.02) + 0.000025 (200 x)$ .

$\frac{0.000025 (4000000 \cdot 0.02 + 200 x)}{100 + x} = y$

Step 2.2

Factor $4$ out of $4000000 \cdot 0.02 + 200 x$ .

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Step 2.2.1

Factor $4$ out of $4000000 \cdot 0.02$ .

$\frac{0.000025 (4 (1000000 \cdot 0.02) + 200 x)}{100 + x} = y$

Step 2.2.2

Factor $4$ out of $200 x$ .

$\frac{0.000025 (4 (1000000 \cdot 0.02) + 4 (50 x))}{100 + x} = y$

Step 2.2.3

Factor $4$ out of $4 (1000000 \cdot 0.02) + 4 (50 x)$ .

$\frac{0.000025 (4 (1000000 \cdot 0.02 + 50 x))}{100 + x} = y$

Step 2.3

Multiply $1000000$ by $0.02$ .

$\frac{0.000025 (4 (20000 + 50 x))}{100 + x} = y$

Step 2.4

Factor.

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Step 2.4.1

Factor $50$ out of $20000 + 50 x$ .

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Step 2.4.1.1

Factor $50$ out of $20000$ .

$\frac{0.000025 (4 (50 \cdot 400 + 50 x))}{100 + x} = y$

Step 2.4.1.2

Factor $50$ out of $50 \cdot 400 + 50 x$ .

$\frac{0.000025 (4 (50 (400 + x)))}{100 + x} = y$

Step 2.4.2

Remove unnecessary parentheses.

$\frac{0.000025 (4 \cdot 50 (400 + x))}{100 + x} = y$

Step 2.5

Factor.

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Step 2.5.1

Multiply $4$ by $50$ .

$\frac{0.000025 (200 (400 + x))}{100 + x} = y$

Step 2.5.2

Remove unnecessary parentheses.

$\frac{0.000025 \cdot 200 (400 + x)}{100 + x} = y$

Step 2.6

Multiply $0.000025$ by $200$ .

$\frac{0.005 (400 + x)}{100 + x} = y$

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.

$100 + x, 1$

Step 3.2

Remove parentheses.

$100 + x, 1$

Step 3.3

The LCM of one and any expression is the expression.

$100 + x$

Step 4

Multiply each term in $\frac{0.005 (400 + x)}{100 + x} = y$ by $100 + x$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{0.005 (400 + x)}{100 + x} = y$ by $100 + x$ .

$\frac{0.005 (400 + x)}{100 + x} (100 + x) = y (100 + x)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.005$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.005 (400 + x)}{100 + x} (100 + x) = y (100 + x)$

Step 4.2.1.2

Rewrite the expression.

$\frac{400 + x}{100 + x} \cdot \frac{100 + x}{200} = y (100 + x)$

Step 4.2.2

Cancel the common factor of $100 + x$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{400 + x}{100 + x} \cdot \frac{100 + x}{200} = y (100 + x)$

Step 4.2.2.2

Rewrite the expression.

$(400 + x) \frac{1}{200} = y (100 + x)$

Step 4.2.3

Apply the distributive property.

$400 (\frac{1}{200}) + x \frac{1}{200} = y (100 + x)$

Step 4.2.4

Cancel the common factor of $200$ .

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Step 4.2.4.1

Factor $200$ out of $400$ .

$200 (2) \frac{1}{200} + x \frac{1}{200} = y (100 + x)$

Step 4.2.4.2

Cancel the common factor.

$200 \cdot 2 \frac{1}{200} + x \frac{1}{200} = y (100 + x)$

Step 4.2.4.3

Rewrite the expression.

$2 + x \frac{1}{200} = y (100 + x)$

Step 4.2.5

Combine $x$ and $\frac{1}{200}$ .

$2 + \frac{x}{200} = y (100 + x)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$2 + \frac{x}{200} = y \cdot 100 + y x$

Step 4.3.2

Move $100$ to the left of $y$ .

$2 + \frac{x}{200} = 100 y + y x$

Step 5

Solve the equation.

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Step 5.1

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

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Step 5.1.1

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

$2 + \frac{x}{200} - y x = 100 y$

Step 5.1.2

To write $- y x$ as a fraction with a common denominator, multiply by $\frac{200}{200}$ .

$2 + \frac{x}{200} - y x \cdot \frac{200}{200} = 100 y$

Step 5.1.3

Combine $- y x$ and $\frac{200}{200}$ .

$2 + \frac{x}{200} + \frac{- y x \cdot 200}{200} = 100 y$

Step 5.1.4

Combine the numerators over the common denominator.

$2 + \frac{x - y x \cdot 200}{200} = 100 y$

Step 5.1.5

Simplify the numerator.

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Step 5.1.5.1

Factor $x$ out of $x - y x \cdot 200$ .

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Step 5.1.5.1.1

Raise $x$ to the power of $1$ .

$2 + \frac{x^{1} - y x \cdot 200}{200} = 100 y$

Step 5.1.5.1.2

Factor $x$ out of $x^{1}$ .

$2 + \frac{x \cdot 1 - y x \cdot 200}{200} = 100 y$

Step 5.1.5.1.3

Factor $x$ out of $- y x \cdot 200$ .

$2 + \frac{x \cdot 1 + x (- y \cdot 200)}{200} = 100 y$

Step 5.1.5.1.4

Factor $x$ out of $x \cdot 1 + x (- y \cdot 200)$ .

$2 + \frac{x (1 - y \cdot 200)}{200} = 100 y$

Step 5.1.5.2

Multiply $200$ by $- 1$ .

$2 + \frac{x (1 - 200 y)}{200} = 100 y$

Step 5.1.6

To write $2$ as a fraction with a common denominator, multiply by $\frac{200}{200}$ .

$2 \cdot \frac{200}{200} + \frac{x (1 - 200 y)}{200} = 100 y$

Step 5.1.7

Combine $2$ and $\frac{200}{200}$ .

$\frac{2 \cdot 200}{200} + \frac{x (1 - 200 y)}{200} = 100 y$

Step 5.1.8

Combine the numerators over the common denominator.

$\frac{2 \cdot 200 + x (1 - 200 y)}{200} = 100 y$

Step 5.1.9

Simplify the numerator.

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Step 5.1.9.1

Multiply $2$ by $200$ .

$\frac{400 + x (1 - 200 y)}{200} = 100 y$

Step 5.1.9.2

Apply the distributive property.

$\frac{400 + x \cdot 1 + x (- 200 y)}{200} = 100 y$

Step 5.1.9.3

Multiply $x$ by $1$ .

$\frac{400 + x + x (- 200 y)}{200} = 100 y$

Step 5.1.9.4

Rewrite using the commutative property of multiplication.

$\frac{400 + x - 200 x y}{200} = 100 y$

Step 5.2

Multiply both sides by $200$ .

$\frac{400 + x - 200 x y}{200} \cdot 200 = 100 y \cdot 200$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{400 + x - 200 x y}{200} \cdot 200$ .

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Step 5.3.1.1.1

Cancel the common factor of $200$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{400 + x - 200 x y}{200} \cdot 200 = 100 y \cdot 200$

Step 5.3.1.1.1.2

Rewrite the expression.

$400 + x - 200 x y = 100 y \cdot 200$

Step 5.3.1.1.2

Simplify the expression.

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Step 5.3.1.1.2.1

Move $400$ .

$x - 200 x y + 400 = 100 y \cdot 200$

Step 5.3.1.1.2.2

Reorder $x$ and $- 200 x y$ .

$- 200 x y + x + 400 = 100 y \cdot 200$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $200$ by $100$ .

$- 200 x y + x + 400 = 20000 y$

Step 5.4

Solve for $x$ .

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Step 5.4.1

Subtract $400$ from both sides of the equation.

$- 200 x y + x = 20000 y - 400$

Step 5.4.2

Factor $x$ out of $- 200 x y + x$ .

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Step 5.4.2.1

Factor $x$ out of $- 200 x y$ .

$x (- 200 y) + x = 20000 y - 400$

Step 5.4.2.2

Raise $x$ to the power of $1$ .

$x (- 200 y) + x = 20000 y - 400$

Step 5.4.2.3

Factor $x$ out of $x^{1}$ .

$x (- 200 y) + x \cdot 1 = 20000 y - 400$

Step 5.4.2.4

Factor $x$ out of $x (- 200 y) + x \cdot 1$ .

$x (- 200 y + 1) = 20000 y - 400$

Step 5.4.3

Divide each term in $x (- 200 y + 1) = 20000 y - 400$ by $- 200 y + 1$ and simplify.

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Step 5.4.3.1

Divide each term in $x (- 200 y + 1) = 20000 y - 400$ by $- 200 y + 1$ .

$\frac{x (- 200 y + 1)}{- 200 y + 1} = \frac{20000 y}{- 200 y + 1} + \frac{- 400}{- 200 y + 1}$

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $- 200 y + 1$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

$\frac{x (- 200 y + 1)}{- 200 y + 1} = \frac{20000 y}{- 200 y + 1} + \frac{- 400}{- 200 y + 1}$

Step 5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{20000 y}{- 200 y + 1} + \frac{- 400}{- 200 y + 1}$

Step 5.4.3.3

Simplify the right side.

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Step 5.4.3.3.1

Combine the numerators over the common denominator.

$x = \frac{20000 y - 400}{- 200 y + 1}$

Step 5.4.3.3.2

Factor $400$ out of $20000 y - 400$ .

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Step 5.4.3.3.2.1

Factor $400$ out of $20000 y$ .

$x = \frac{400 (50 y) - 400}{- 200 y + 1}$

Step 5.4.3.3.2.2

Factor $400$ out of $- 400$ .

$x = \frac{400 (50 y) + 400 (- 1)}{- 200 y + 1}$

Step 5.4.3.3.2.3

Factor $400$ out of $400 (50 y) + 400 (- 1)$ .

$x = \frac{400 (50 y - 1)}{- 200 y + 1}$

Step 5.4.3.3.3

Factor $- 1$ out of $- 200 y$ .

$x = \frac{400 (50 y - 1)}{- (200 y) + 1}$

Step 5.4.3.3.4

Rewrite $1$ as $- 1 (- 1)$ .

$x = \frac{400 (50 y - 1)}{- (200 y) - 1 (- 1)}$

Step 5.4.3.3.5

Factor $- 1$ out of $- (200 y) - 1 (- 1)$ .

$x = \frac{400 (50 y - 1)}{- (200 y - 1)}$

Step 5.4.3.3.6

Rewrite negatives.

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Step 5.4.3.3.6.1

Rewrite $- (200 y - 1)$ as $- 1 (200 y - 1)$ .

$x = \frac{400 (50 y - 1)}{- 1 (200 y - 1)}$

Step 5.4.3.3.6.2

Move the negative in front of the fraction.

$x = - \frac{400 (50 y - 1)}{200 y - 1}$