Finite Math Examples

$x^{2} - 4 y^{2} = 5$ , $x^{2} - 2 x y = 15$

Step 1

Solve for $y$ in $x^{2} - 2 x y = 15$ .

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Step 1.1

Subtract $x^{2}$ from both sides of the equation.

$- 2 x y = 15 - x^{2}$

$x^{2} - 4 y^{2} = 5$

Step 1.2

Divide each term in $- 2 x y = 15 - x^{2}$ by $- 2 x$ and simplify.

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Step 1.2.1

Divide each term in $- 2 x y = 15 - x^{2}$ by $- 2 x$ .

$\frac{- 2 x y}{- 2 x} = \frac{15}{- 2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x y}{- 2 x} = \frac{15}{- 2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.2.1.2

Rewrite the expression.

$\frac{x y}{x} = \frac{15}{- 2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

$\frac{x y}{x} = \frac{15}{- 2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.2.2

Cancel the common factor of $x$ .

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Step 1.2.2.2.1

Cancel the common factor.

$\frac{x y}{x} = \frac{15}{- 2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{15}{- 2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

$y = \frac{15}{- 2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

$y = \frac{15}{- 2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{15}{2 x} + \frac{- x^{2}}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.3.1.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.2.3.1.2.1

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

$y = - \frac{15}{2 x} + \frac{x (- x)}{- 2 x}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $x$ out of $- 2 x$ .

$y = - \frac{15}{2 x} + \frac{x (- x)}{x \cdot - 2}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{15}{2 x} + \frac{x (- x)}{x \cdot - 2}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{15}{2 x} + \frac{- x}{- 2}$

$x^{2} - 4 y^{2} = 5$

$y = - \frac{15}{2 x} + \frac{- x}{- 2}$

$x^{2} - 4 y^{2} = 5$

$y = - \frac{15}{2 x} + \frac{- x}{- 2}$

$x^{2} - 4 y^{2} = 5$

Step 1.2.3.1.3

Dividing two negative values results in a positive value.

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 y^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 y^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 y^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 y^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 y^{2} = 5$

Step 2

Replace all occurrences of $y$ with $- \frac{15}{2 x} + \frac{x}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} - 4 y^{2} = 5$ with $- \frac{15}{2 x} + \frac{x}{2}$ .

$x^{2} - 4 {(- \frac{15}{2 x} + \frac{x}{2})}^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} - 4 {(- \frac{15}{2 x} + \frac{x}{2})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(- \frac{15}{2 x} + \frac{x}{2})}^{2}$ as $(- \frac{15}{2 x} + \frac{x}{2}) (- \frac{15}{2 x} + \frac{x}{2})$ .

$x^{2} - 4 ((- \frac{15}{2 x} + \frac{x}{2}) (- \frac{15}{2 x} + \frac{x}{2})) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.2

Expand $(- \frac{15}{2 x} + \frac{x}{2}) (- \frac{15}{2 x} + \frac{x}{2})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} - 4 (- \frac{15}{2 x} \cdot (- \frac{15}{2 x} + \frac{x}{2}) + \frac{x}{2} \cdot (- \frac{15}{2 x} + \frac{x}{2})) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} - 4 (- \frac{15}{2 x} \cdot (- \frac{15}{2 x}) - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x} + \frac{x}{2})) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} - 4 (- \frac{15}{2 x} \cdot (- \frac{15}{2 x}) - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (- \frac{15}{2 x} \cdot (- \frac{15}{2 x}) - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $- \frac{15}{2 x} (- \frac{15}{2 x})$ .

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Step 2.2.1.1.3.1.1.1

Multiply $- 1$ by $- 1$ .

$x^{2} - 4 (1 (\frac{15}{2 x}) \cdot \frac{15}{2 x} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.1.2

Multiply $\frac{15}{2 x}$ by $1$ .

$x^{2} - 4 (\frac{15}{2 x} \cdot \frac{15}{2 x} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.1.3

Multiply $\frac{15}{2 x}$ by $\frac{15}{2 x}$ .

$x^{2} - 4 (\frac{15 \cdot 15}{2 x (2 x)} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.1.4

Multiply $15$ by $15$ .

$x^{2} - 4 (\frac{225}{2 x (2 x)} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.1.5

Multiply $2$ by $2$ .

$x^{2} - 4 (\frac{225}{4 x \cdot x} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.1.6

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

$x^{2} - 4 (\frac{225}{4 (x \cdot x)} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.1.7

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

$x^{2} - 4 (\frac{225}{4 (x \cdot x)} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.1.8

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

$x^{2} - 4 (\frac{225}{4 x^{1 + 1}} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.1.9

Add $1$ and $1$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.2

Cancel the common factor of $x$ .

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Step 2.2.1.1.3.1.2.1

Move the leading negative in $- \frac{15}{2 x}$ into the numerator.

$x^{2} - 4 (\frac{225}{4 x^{2}} + \frac{- 15}{2 x} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.2.2

Factor $x$ out of $2 x$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} + \frac{- 15}{x \cdot 2} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.2.3

Cancel the common factor.

$x^{2} - 4 (\frac{225}{4 x^{2}} + \frac{- 15}{x \cdot 2} \cdot \frac{x}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.2.4

Rewrite the expression.

$x^{2} - 4 (\frac{225}{4 x^{2}} + \frac{- 15}{2} \cdot \frac{1}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (\frac{225}{4 x^{2}} + \frac{- 15}{2} \cdot \frac{1}{2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.3

Multiply $\frac{- 15}{2}$ by $\frac{1}{2}$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} + \frac{- 15}{2 \cdot 2} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.4

Multiply $2$ by $2$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} + \frac{- 15}{4} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.5

Move the negative in front of the fraction.

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{x}{2} \cdot (- \frac{15}{2 x}) + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.6

Rewrite using the commutative property of multiplication.

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{x}{2} \cdot \frac{15}{2 x} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.7

Cancel the common factor of $x$ .

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Step 2.2.1.1.3.1.7.1

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

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{- x}{2} \cdot \frac{15}{2 x} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.7.2

Factor $x$ out of $- x$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{x \cdot - 1}{2} \cdot \frac{15}{2 x} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.7.3

Factor $x$ out of $2 x$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{x \cdot - 1}{2} \cdot \frac{15}{x \cdot 2} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.7.4

Cancel the common factor.

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{x \cdot - 1}{2} \cdot \frac{15}{x \cdot 2} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.7.5

Rewrite the expression.

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{- 1}{2} \cdot \frac{15}{2} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{- 1}{2} \cdot \frac{15}{2} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.8

Multiply $\frac{- 1}{2}$ by $\frac{15}{2}$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{- 1 \cdot 15}{2 \cdot 2} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.9

Multiply $- 1$ by $15$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{- 15}{2 \cdot 2} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.10

Multiply $2$ by $2$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} + \frac{- 15}{4} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.11

Move the negative in front of the fraction.

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x}{2} \cdot \frac{x}{2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.12

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

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Step 2.2.1.1.3.1.12.1

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

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x \cdot x}{2 \cdot 2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.12.2

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

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x \cdot x}{2 \cdot 2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.12.3

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

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x \cdot x}{2 \cdot 2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.12.4

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

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x^{1 + 1}}{2 \cdot 2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.12.5

Add $1$ and $1$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x^{2}}{2 \cdot 2}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.1.12.6

Multiply $2$ by $2$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{4} - \frac{15}{4} + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.3.2

Subtract $\frac{15}{4}$ from $- \frac{15}{4}$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - 2 (\frac{15}{4}) + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (\frac{225}{4 x^{2}} - 2 (\frac{15}{4}) + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1.1

Factor $2$ out of $- 2$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} + 2 (- 1) (\frac{15}{4}) + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.4.1.2

Factor $2$ out of $4$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} + 2 \cdot (- 1 \frac{15}{2 \cdot 2}) + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.4.1.3

Cancel the common factor.

$x^{2} - 4 (\frac{225}{4 x^{2}} + 2 \cdot (- 1 \frac{15}{2 \cdot 2}) + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.4.1.4

Rewrite the expression.

$x^{2} - 4 (\frac{225}{4 x^{2}} - 1 (\frac{15}{2}) + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (\frac{225}{4 x^{2}} - 1 (\frac{15}{2}) + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.4.2

Rewrite $- 1 (\frac{15}{2})$ as $- (\frac{15}{2})$ .

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{2} + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 4 (\frac{225}{4 x^{2}} - \frac{15}{2} + \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.5

Apply the distributive property.

$x^{2} - 4 \frac{225}{4 x^{2}} - 4 (- \frac{15}{2}) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6

Simplify.

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Step 2.2.1.1.6.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.6.1.1

Factor $4$ out of $- 4$ .

$x^{2} + 4 (- 1) (\frac{225}{4 x^{2}}) - 4 (- \frac{15}{2}) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.1.2

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

$x^{2} + 4 (- 1) (\frac{225}{4 (x^{2})}) - 4 (- \frac{15}{2}) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.1.3

Cancel the common factor.

$x^{2} + 4 \cdot (- 1 \frac{225}{4 x^{2}}) - 4 (- \frac{15}{2}) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.1.4

Rewrite the expression.

$x^{2} - 1 \frac{225}{x^{2}} - 4 (- \frac{15}{2}) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 1 \frac{225}{x^{2}} - 4 (- \frac{15}{2}) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.6.2.1

Move the leading negative in $- \frac{15}{2}$ into the numerator.

$x^{2} - 1 \frac{225}{x^{2}} - 4 (\frac{- 15}{2}) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.2.2

Factor $2$ out of $- 4$ .

$x^{2} - 1 \frac{225}{x^{2}} + 2 (- 2) (\frac{- 15}{2}) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.2.3

Cancel the common factor.

$x^{2} - 1 \frac{225}{x^{2}} + 2 \cdot (- 2 (\frac{- 15}{2})) - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.2.4

Rewrite the expression.

$x^{2} - 1 \frac{225}{x^{2}} - 2 \cdot - 15 - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 1 \frac{225}{x^{2}} - 2 \cdot - 15 - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.3

Multiply $- 2$ by $- 15$ .

$x^{2} - 1 \frac{225}{x^{2}} + 30 - 4 \frac{x^{2}}{4} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.4

Cancel the common factor of $4$ .

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Step 2.2.1.1.6.4.1

Factor $4$ out of $- 4$ .

$x^{2} - 1 \frac{225}{x^{2}} + 30 + 4 (- 1) (\frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.4.2

Cancel the common factor.

$x^{2} - 1 \frac{225}{x^{2}} + 30 + 4 \cdot (- 1 \frac{x^{2}}{4}) = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.6.4.3

Rewrite the expression.

$x^{2} - 1 \frac{225}{x^{2}} + 30 - 1 x^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 1 \frac{225}{x^{2}} + 30 - 1 x^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - 1 \frac{225}{x^{2}} + 30 - 1 x^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.7

Simplify each term.

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Step 2.2.1.1.7.1

Rewrite $- 1 \frac{225}{x^{2}}$ as $- \frac{225}{x^{2}}$ .

$x^{2} - \frac{225}{x^{2}} + 30 - 1 x^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.1.7.2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$x^{2} - \frac{225}{x^{2}} + 30 - x^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - \frac{225}{x^{2}} + 30 - x^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} - \frac{225}{x^{2}} + 30 - x^{2} = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.2

Combine the opposite terms in $x^{2} - \frac{225}{x^{2}} + 30 - x^{2}$ .

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Step 2.2.1.2.1

Subtract $x^{2}$ from $x^{2}$ .

$- \frac{225}{x^{2}} + 30 + 0 = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 2.2.1.2.2

Add $- \frac{225}{x^{2}} + 30$ and $0$ .

$- \frac{225}{x^{2}} + 30 = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$- \frac{225}{x^{2}} + 30 = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$- \frac{225}{x^{2}} + 30 = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$- \frac{225}{x^{2}} + 30 = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$- \frac{225}{x^{2}} + 30 = 5$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3

Solve for $x$ in $- \frac{225}{x^{2}} + 30 = 5$ .

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Step 3.1

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

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Step 3.1.1

Subtract $30$ from both sides of the equation.

$- \frac{225}{x^{2}} = 5 - 30$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.1.2

Subtract $30$ from $5$ .

$- \frac{225}{x^{2}} = - 25$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$- \frac{225}{x^{2}} = - 25$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$x^{2}, 1$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.2.2

The LCM of one and any expression is the expression.

$x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.3

Multiply each term in $- \frac{225}{x^{2}} = - 25$ by $x^{2}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $- \frac{225}{x^{2}} = - 25$ by $x^{2}$ .

$- \frac{225}{x^{2}} \cdot x^{2} = - 25 x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $x^{2}$ .

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Step 3.3.2.1.1

Move the leading negative in $- \frac{225}{x^{2}}$ into the numerator.

$\frac{- 225}{x^{2}} \cdot x^{2} = - 25 x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.3.2.1.2

Cancel the common factor.

$\frac{- 225}{x^{2}} \cdot x^{2} = - 25 x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.3.2.1.3

Rewrite the expression.

$- 225 = - 25 x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$- 225 = - 25 x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$- 225 = - 25 x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$- 225 = - 25 x^{2}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $- 25 x^{2} = - 225$ .

$- 25 x^{2} = - 225$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.2

Divide each term in $- 25 x^{2} = - 225$ by $- 25$ and simplify.

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Step 3.4.2.1

Divide each term in $- 25 x^{2} = - 225$ by $- 25$ .

$\frac{- 25 x^{2}}{- 25} = \frac{- 225}{- 25}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $- 25$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{- 25 x^{2}}{- 25} = \frac{- 225}{- 25}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.2.2.1.2

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

$x^{2} = \frac{- 225}{- 25}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} = \frac{- 225}{- 25}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} = \frac{- 225}{- 25}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $- 225$ by $- 25$ .

$x^{2} = 9$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} = 9$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x^{2} = 9$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.3

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

$x = \pm \sqrt{9}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.4

Simplify $\pm \sqrt{9}$ .

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Step 3.4.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x = \pm 3$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.5

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

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Step 3.4.5.1

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

$x = 3$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.5.2

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

$x = - 3$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 3.4.5.3

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

$x = 3, - 3$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x = 3, - 3$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x = 3, - 3$

$y = - \frac{15}{2 x} + \frac{x}{2}$

$x = 3, - 3$

$y = - \frac{15}{2 x} + \frac{x}{2}$

Step 4

Replace all occurrences of $x$ with $3$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - \frac{15}{2 x} + \frac{x}{2}$ with $3$ .

$y = - \frac{15}{2 (3)} + \frac{3}{2}$

$x = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{15}{2 (3)} + \frac{3}{2}$ .

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Step 4.2.1.1

Cancel the common factor of $15$ and $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $15$ .

$y = - \frac{3 \cdot 5}{2 \cdot 3} + \frac{3}{2}$

$x = 3$

Step 4.2.1.1.2

Cancel the common factors.

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Step 4.2.1.1.2.1

Factor $3$ out of $2 \cdot 3$ .

$y = - \frac{3 \cdot 5}{3 \cdot 2} + \frac{3}{2}$

$x = 3$

Step 4.2.1.1.2.2

Cancel the common factor.

$y = - \frac{3 \cdot 5}{3 \cdot 2} + \frac{3}{2}$

$x = 3$

Step 4.2.1.1.2.3

Rewrite the expression.

$y = - \frac{5}{2} + \frac{3}{2}$

$x = 3$

$y = - \frac{5}{2} + \frac{3}{2}$

$x = 3$

$y = - \frac{5}{2} + \frac{3}{2}$

$x = 3$

Step 4.2.1.2

Combine the numerators over the common denominator.

$y = \frac{- 5 + 3}{2}$

$x = 3$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Add $- 5$ and $3$ .

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

$x = 3$

Step 4.2.1.3.2

Divide $- 2$ by $2$ .

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

Step 5

Replace all occurrences of $x$ with $- 3$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - \frac{15}{2 x} + \frac{x}{2}$ with $- 3$ .

$y = - \frac{15}{2 (- 3)} + \frac{- 3}{2}$

$x = - 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{15}{2 (- 3)} + \frac{- 3}{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Cancel the common factor of $15$ and $- 3$ .

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Step 5.2.1.1.1.1

Factor $3$ out of $15$ .

$y = - \frac{3 (5)}{2 (- 3)} + \frac{- 3}{2}$

$x = - 3$

Step 5.2.1.1.1.2

Cancel the common factors.

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Step 5.2.1.1.1.2.1

Factor $3$ out of $2 (- 3)$ .

$y = - \frac{3 (5)}{3 (2 \cdot (- 1))} + \frac{- 3}{2}$

$x = - 3$

Step 5.2.1.1.1.2.2

Cancel the common factor.

$y = - \frac{3 \cdot 5}{3 (2 \cdot (- 1))} + \frac{- 3}{2}$

$x = - 3$

Step 5.2.1.1.1.2.3

Rewrite the expression.

$y = - \frac{5}{2 \cdot (- 1)} + \frac{- 3}{2}$

$x = - 3$

$y = - \frac{5}{2 \cdot (- 1)} + \frac{- 3}{2}$

$x = - 3$

$y = - \frac{5}{2 \cdot (- 1)} + \frac{- 3}{2}$

$x = - 3$

Step 5.2.1.1.2

Multiply $2$ by $- 1$ .

$y = - \frac{5}{- 2} + \frac{- 3}{2}$

$x = - 3$

Step 5.2.1.1.3

Move the negative in front of the fraction.

$y = \frac{5}{2} + \frac{- 3}{2}$

$x = - 3$

Step 5.2.1.1.4

Multiply $- - \frac{5}{2}$ .

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Step 5.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$y = 1 (\frac{5}{2}) + \frac{- 3}{2}$

$x = - 3$

Step 5.2.1.1.4.2

Multiply $\frac{5}{2}$ by $1$ .

$y = \frac{5}{2} + \frac{- 3}{2}$

$x = - 3$

$y = \frac{5}{2} + \frac{- 3}{2}$

$x = - 3$

Step 5.2.1.1.5

Move the negative in front of the fraction.

$y = \frac{5}{2} - \frac{3}{2}$

$x = - 3$

$y = \frac{5}{2} - \frac{3}{2}$

$x = - 3$

Step 5.2.1.2

Combine fractions.

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Step 5.2.1.2.1

Combine the numerators over the common denominator.

$y = \frac{5 - 3}{2}$

$x = - 3$

Step 5.2.1.2.2

Simplify the expression.

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Step 5.2.1.2.2.1

Subtract $3$ from $5$ .

$y = \frac{2}{2}$

$x = - 3$

Step 5.2.1.2.2.2

Divide $2$ by $2$ .

$y = 1$

$x = - 3$

$y = 1$

$x = - 3$

$y = 1$

$x = - 3$

$y = 1$

$x = - 3$

$y = 1$

$x = - 3$

$y = 1$

$x = - 3$

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(3, - 1)$

$(- 3, 1)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(3, - 1), (- 3, 1)$

Equation Form:

$x = 3, y = - 1$

$x = - 3, y = 1$

Step 8