Precalculus Examples

$y = \frac{\frac{4}{m}}{\frac{1}{m} + \frac{2}{x}}$ , $m = \frac{1}{2 x}$

Step 1

Solve the equation for $m$ .

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Rewrite the equation as $\frac{\frac{4}{m}}{\frac{1}{m} + \frac{2}{x}} = y$ .

$\frac{\frac{4}{m}}{\frac{1}{m} + \frac{2}{x}} = y$

$m = \frac{1}{2 x}$

Factor each term.

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Multiply the numerator by the reciprocal of the denominator.

$\frac{4}{m} \cdot \frac{1}{\frac{1}{m} + \frac{2}{x}} = y$

$m = \frac{1}{2 x}$

Simplify the denominator.

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To write $\frac{1}{m}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4}{m} \cdot \frac{1}{\frac{1}{m} \cdot \frac{x}{x} + \frac{2}{x}} = y$

$m = \frac{1}{2 x}$

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

$\frac{4}{m} \cdot \frac{1}{\frac{1}{m} \cdot \frac{x}{x} + \frac{2}{x} \cdot \frac{m}{m}} = y$

$m = \frac{1}{2 x}$

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

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Multiply $\frac{1}{m}$ by $\frac{x}{x}$ .

$\frac{4}{m} \cdot \frac{1}{\frac{x}{m x} + \frac{2}{x} \cdot \frac{m}{m}} = y$

$m = \frac{1}{2 x}$

Multiply $\frac{2}{x}$ by $\frac{m}{m}$ .

$\frac{4}{m} \cdot \frac{1}{\frac{x}{m x} + \frac{2 m}{x m}} = y$

$m = \frac{1}{2 x}$

Reorder the factors of $x m$ .

$\frac{4}{m} \cdot \frac{1}{\frac{x}{m x} + \frac{2 m}{m x}} = y$

$m = \frac{1}{2 x}$

$\frac{4}{m} \cdot \frac{1}{\frac{x}{m x} + \frac{2 m}{m x}} = y$

$m = \frac{1}{2 x}$

Combine the numerators over the common denominator.

$\frac{4}{m} \cdot \frac{1}{\frac{x + 2 m}{m x}} = y$

$m = \frac{1}{2 x}$

$\frac{4}{m} \cdot \frac{1}{\frac{x + 2 m}{m x}} = y$

$m = \frac{1}{2 x}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{4}{m} \cdot (1 (\frac{m x}{x + 2 m})) = y$

$m = \frac{1}{2 x}$

Multiply $\frac{m x}{x + 2 m}$ by $1$ .

$\frac{4}{m} \cdot \frac{m x}{x + 2 m} = y$

$m = \frac{1}{2 x}$

Cancel the common factor of $m$ .

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Factor $m$ out of $m x$ .

$\frac{4}{m} \cdot \frac{m (x)}{x + 2 m} = y$

$m = \frac{1}{2 x}$

Cancel the common factor.

$\frac{4}{m} \cdot \frac{m x}{x + 2 m} = y$

$m = \frac{1}{2 x}$

Rewrite the expression.

$4 (\frac{x}{x + 2 m}) = y$

$m = \frac{1}{2 x}$

$4 (\frac{x}{x + 2 m}) = y$

$m = \frac{1}{2 x}$

Combine $4$ and $\frac{x}{x + 2 m}$ .

$\frac{4 x}{x + 2 m} = y$

$m = \frac{1}{2 x}$

$\frac{4 x}{x + 2 m} = y$

$m = \frac{1}{2 x}$

Find the LCD of the terms in the equation.

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

$x + 2 m, 1$

$m = \frac{1}{2 x}$

Remove parentheses.

$x + 2 m, 1$

$m = \frac{1}{2 x}$

The LCM of one and any expression is the expression.

$x + 2 m$

$m = \frac{1}{2 x}$

$x + 2 m$

$m = \frac{1}{2 x}$

Multiply each term in $\frac{4 x}{x + 2 m} = y$ by $x + 2 m$ to eliminate the fractions.

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Multiply each term in $\frac{4 x}{x + 2 m} = y$ by $x + 2 m$ .

$\frac{4 x}{x + 2 m} \cdot (x + 2 m) = y (x + 2 m)$

$m = \frac{1}{2 x}$

Simplify the left side.

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Cancel the common factor of $x + 2 m$ .

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Cancel the common factor.

$\frac{4 x}{x + 2 m} \cdot (x + 2 m) = y (x + 2 m)$

$m = \frac{1}{2 x}$

Rewrite the expression.

$4 x = y (x + 2 m)$

$m = \frac{1}{2 x}$

$4 x = y (x + 2 m)$

$m = \frac{1}{2 x}$

$4 x = y (x + 2 m)$

$m = \frac{1}{2 x}$

Simplify the right side.

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Apply the distributive property.

$4 x = y x + y (2 m)$

$m = \frac{1}{2 x}$

Rewrite using the commutative property of multiplication.

$4 x = y x + 2 y m$

$m = \frac{1}{2 x}$

$4 x = y x + 2 y m$

$m = \frac{1}{2 x}$

$4 x = y x + 2 y m$

$m = \frac{1}{2 x}$

Solve the equation.

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Rewrite the equation as $y x + 2 y m = 4 x$ .

$y x + 2 y m = 4 x$

$m = \frac{1}{2 x}$

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

$2 y m = 4 x - y x$

$m = \frac{1}{2 x}$

Divide each term in $2 y m = 4 x - y x$ by $2 y$ and simplify.

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Divide each term in $2 y m = 4 x - y x$ by $2 y$ .

$\frac{2 y m}{2 y} = \frac{4 x}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 y m}{2 y} = \frac{4 x}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Rewrite the expression.

$\frac{y m}{y} = \frac{4 x}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

$\frac{y m}{y} = \frac{4 x}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Cancel the common factor of $y$ .

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Cancel the common factor.

$\frac{y m}{y} = \frac{4 x}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Divide $m$ by $1$ .

$m = \frac{4 x}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

$m = \frac{4 x}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

$m = \frac{4 x}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Simplify the right side.

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Simplify each term.

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Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4 x$ .

$m = \frac{2 (2 x)}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Cancel the common factors.

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Factor $2$ out of $2 y$ .

$m = \frac{2 (2 x)}{2 (y)} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Cancel the common factor.

$m = \frac{2 (2 x)}{2 y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Rewrite the expression.

$m = \frac{2 x}{y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

$m = \frac{2 x}{y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

$m = \frac{2 x}{y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Cancel the common factor of $y$ .

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Cancel the common factor.

$m = \frac{2 x}{y} + \frac{- y x}{2 y}$

$m = \frac{1}{2 x}$

Rewrite the expression.

$m = \frac{2 x}{y} + \frac{- 1 x}{2}$

$m = \frac{1}{2 x}$

$m = \frac{2 x}{y} + \frac{- 1 x}{2}$

$m = \frac{1}{2 x}$

Move the negative in front of the fraction.

$m = \frac{2 x}{y} - \frac{x}{2}$

$m = \frac{1}{2 x}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$m = \frac{1}{2 x}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$m = \frac{1}{2 x}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$m = \frac{1}{2 x}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$m = \frac{1}{2 x}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$m = \frac{1}{2 x}$

Step 2

Solve the equation for $x$ .

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Find the LCD of the terms in the equation.

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

$y, 2, 2 x$

$m = \frac{2 x}{y} - \frac{x}{2}$

Since $y, 2, 2 x$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 2, 2$ then find LCM for the variable part $y^{1}, x^{1}$ .

Since $y, 2, 2 x$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 2, 2$ then find LCM for the variable part y,x.

$m = \frac{2 x}{y} - \frac{x}{2}$

The LCM is the smallest positive number that all of the numbers divide into evenly.

$1$ List the prime factors of each number.

$2$ Multiply each factor the greatest number of times it occurs in either number.

$m = \frac{2 x}{y} - \frac{x}{2}$

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

The LCM of $1, 2, 2$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

$m = \frac{2 x}{y} - \frac{x}{2}$

The factor for $y^{1}$ is $y$ itself.

$y = y$

y occurs $1$ time.

$m = \frac{2 x}{y} - \frac{x}{2}$

The factor for $x^{1}$ is $x$ itself.

$x = x$

x occurs $1$ time.

$m = \frac{2 x}{y} - \frac{x}{2}$

The LCM of $y^{1}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y \cdot x$

$m = \frac{2 x}{y} - \frac{x}{2}$

Multiply $y$ by $x$ .

$y x$

$m = \frac{2 x}{y} - \frac{x}{2}$

The LCM for $y, 2, 2 x$ is the numeric part $2$ multiplied by the variable part.

$2 y x$

$m = \frac{2 x}{y} - \frac{x}{2}$

$2 y x$

$m = \frac{2 x}{y} - \frac{x}{2}$

Multiply each term in $\frac{2 x}{y} - \frac{x}{2} = \frac{1}{2 x}$ by $2 y x$ to eliminate the fractions.

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Multiply each term in $\frac{2 x}{y} - \frac{x}{2} = \frac{1}{2 x}$ by $2 y x$ .

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

$m = \frac{2 x}{y} - \frac{x}{2}$

Simplify the left side.

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Simplify each term.

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Rewrite using the commutative property of multiplication.

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

$m = \frac{2 x}{y} - \frac{x}{2}$

Multiply $2 \frac{2 x}{y}$ .

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Combine $2$ and $\frac{2 x}{y}$ .

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

$m = \frac{2 x}{y} - \frac{x}{2}$

Multiply $2$ by $2$ .

$\frac{4 x}{y} \cdot (y x) - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

$\frac{4 x}{y} \cdot (y x) - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor of $y$ .

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Factor $y$ out of $y x$ .

$\frac{4 x}{y} \cdot (y (x)) - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor.

$\frac{4 x}{y} \cdot (y x) - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Rewrite the expression.

$4 x \cdot x - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

$4 x \cdot x - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

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

$4 (x \cdot x) - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

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

$4 (x \cdot x) - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x^{1 + 1} - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Add $1$ and $1$ .

$4 x^{2} - \frac{x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{x}{2}$ into the numerator.

$4 x^{2} + \frac{- x}{2} \cdot (2 y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Factor $2$ out of $2 y x$ .

$4 x^{2} + \frac{- x}{2} \cdot (2 (y x)) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor.

$4 x^{2} + \frac{- x}{2} \cdot (2 (y x)) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Rewrite the expression.

$4 x^{2} - x (y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

$4 x^{2} - x (y x) = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

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

$4 x^{2} - x \cdot x y = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

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

$4 x^{2} - x \cdot x y = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x^{2} - x^{1 + 1} y = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Add $1$ and $1$ .

$4 x^{2} - x^{2} y = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

$4 x^{2} - x^{2} y = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

$4 x^{2} - x^{2} y = \frac{1}{2 x} \cdot (2 y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Simplify the right side.

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Rewrite using the commutative property of multiplication.

$4 x^{2} - x^{2} y = 2 (\frac{1}{2 x}) \cdot (y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 x$ .

$4 x^{2} - x^{2} y = 2 (\frac{1}{2 (x)}) \cdot (y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor.

$4 x^{2} - x^{2} y = 2 (\frac{1}{2 x}) \cdot (y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Rewrite the expression.

$4 x^{2} - x^{2} y = \frac{1}{x} \cdot (y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

$4 x^{2} - x^{2} y = \frac{1}{x} \cdot (y x)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor of $x$ .

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Factor $x$ out of $y x$ .

$4 x^{2} - x^{2} y = \frac{1}{x} \cdot (x y)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor.

$4 x^{2} - x^{2} y = \frac{1}{x} \cdot (x y)$

$m = \frac{2 x}{y} - \frac{x}{2}$

Rewrite the expression.

$4 x^{2} - x^{2} y = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

$4 x^{2} - x^{2} y = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

$4 x^{2} - x^{2} y = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

$4 x^{2} - x^{2} y = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

Solve the equation.

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Factor $x^{2}$ out of $4 x^{2} - x^{2} y$ .

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Factor $x^{2}$ out of $4 x^{2}$ .

$x^{2} \cdot 4 - x^{2} y = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

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

$x^{2} \cdot 4 + x^{2} (- 1 y) = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

Factor $x^{2}$ out of $x^{2} \cdot 4 + x^{2} (- 1 y)$ .

$x^{2} \cdot (4 - 1 y) = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

Multiply $x^{2}$ by $4 - 1 y$ .

$x^{2} (4 - 1 y) = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x^{2} (4 - 1 y) = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

Rewrite $- 1 y$ as $- y$ .

$x^{2} (4 - y) = y$

$m = \frac{2 x}{y} - \frac{x}{2}$

Divide each term in $x^{2} (4 - y) = y$ by $4 - y$ and simplify.

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Divide each term in $x^{2} (4 - y) = y$ by $4 - y$ .

$\frac{x^{2} (4 - y)}{4 - y} = \frac{y}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Simplify the left side.

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Cancel the common factor of $4 - y$ .

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Cancel the common factor.

$\frac{x^{2} (4 - y)}{4 - y} = \frac{y}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{y}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x^{2} = \frac{y}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x^{2} = \frac{y}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x^{2} = \frac{y}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{y}{4 - y}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Simplify $\pm \sqrt{\frac{y}{4 - y}}$ .

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Rewrite $\sqrt{\frac{y}{4 - y}}$ as $\frac{\sqrt{y}}{\sqrt{4 - y}}$ .

$x = \pm \frac{\sqrt{y}}{\sqrt{4 - y}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Multiply $\frac{\sqrt{y}}{\sqrt{4 - y}}$ by $\frac{\sqrt{4 - y}}{\sqrt{4 - y}}$ .

$x = \pm \frac{\sqrt{y}}{\sqrt{4 - y}} \cdot \frac{\sqrt{4 - y}}{\sqrt{4 - y}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Combine and simplify the denominator.

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Multiply $\frac{\sqrt{y}}{\sqrt{4 - y}}$ by $\frac{\sqrt{4 - y}}{\sqrt{4 - y}}$ .

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{\sqrt{4 - y} \sqrt{4 - y}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Raise $\sqrt{4 - y}$ to the power of $1$ .

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{\sqrt{4 - y} \sqrt{4 - y}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Raise $\sqrt{4 - y}$ to the power of $1$ .

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{\sqrt{4 - y} \sqrt{4 - y}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{{\sqrt{4 - y}}^{1 + 1}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{{\sqrt{4 - y}}^{2}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Rewrite ${\sqrt{4 - y}}^{2}$ as $4 - y$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - y}$ as ${(4 - y)}^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{{({(4 - y)}^{\frac{1}{2}})}^{2}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{{(4 - y)}^{\frac{1}{2} \cdot 2}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{{(4 - y)}^{\frac{2}{2}}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{{(4 - y)}^{\frac{2}{2}}}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Rewrite the expression.

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Simplify.

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x = \pm \frac{\sqrt{y} \sqrt{4 - y}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x = \pm \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 x}{y} - \frac{x}{2}$

Step 3

Simplify the right side.

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Simplify $\frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{y} - \frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{2}$ .

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To write $\frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{y}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$m = \frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{y} \cdot \frac{2}{2} - \frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{2}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

To write $- \frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{2}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$m = \frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{y} \cdot \frac{2}{2} - \frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{2} \cdot \frac{y}{y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Write each expression with a common denominator of $y \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{y}$ by $\frac{2}{2}$ .

$m = \frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) \cdot 2}{y \cdot 2} - \frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{2} \cdot \frac{y}{y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Multiply $\frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})}{2}$ by $\frac{y}{y}$ .

$m = \frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) \cdot 2}{y \cdot 2} - \frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) y}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Reorder the factors of $y \cdot 2$ .

$m = \frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) \cdot 2}{2 y} - \frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) y}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) \cdot 2}{2 y} - \frac{(\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) y}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Combine the numerators over the common denominator.

$m = \frac{2 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) \cdot 2 - (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) y}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Multiply $2$ by $2$ .

$m = \frac{(4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) - \frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{(4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) - \frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$m = \frac{(4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) - \frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}$

Step 4

Simplify the right side.

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Simplify $\frac{(4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) - \frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$ .

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Simplify the numerator.

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To write $4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})$ as a fraction with a common denominator, multiply by $\frac{4 - y}{4 - y}$ .

$m = \frac{(4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) \cdot \frac{4 - y}{4 - y} - \frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Combine $4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y})$ and $\frac{4 - y}{4 - y}$ .

$m = \frac{(\frac{4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) (4 - y)}{4 - y} - \frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Combine the numerators over the common denominator.

$m = \frac{(\frac{4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) (4 - y) - \sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Simplify the numerator.

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Apply the distributive property.

$m = \frac{(\frac{4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) \cdot 4 + 4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) (- y) - \sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Multiply $4$ by $4$ .

$m = \frac{(\frac{16 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) + 4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) (- y) - \sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Multiply $- 1$ by $4$ .

$m = \frac{(\frac{16 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) - 4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) y - \sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

Reorder factors in $\frac{(\frac{16 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) - 4 (\frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}) y - \sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{(4 - y) y})}{2 y}$ .

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = - \frac{\sqrt{y (4 - y)}}{4 - y}$

$x = \frac{\sqrt{y (4 - y)}}{4 - y}, - \frac{\sqrt{y (4 - y)}}{4 - y}$