Finite Math Examples

$x^{2} - 4 y^{2} = 16$ , $2 y - x = 2$

Step 1

Solve for $y$ in $2 y - x = 2$ .

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Step 1.1

Add $x$ to both sides of the equation.

$2 y = 2 + x$

$x^{2} - 4 y^{2} = 16$

Step 1.2

Divide each term in $2 y = 2 + x$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 y = 2 + x$ by $2$ .

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

$x^{2} - 4 y^{2} = 16$

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.

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

$x^{2} - 4 y^{2} = 16$

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

$x^{2} - 4 y^{2} = 16$

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

$x^{2} - 4 y^{2} = 16$

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

$x^{2} - 4 y^{2} = 16$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $2$ by $2$ .

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

$x^{2} - 4 y^{2} = 16$

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

$x^{2} - 4 y^{2} = 16$

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

$x^{2} - 4 y^{2} = 16$

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

$x^{2} - 4 y^{2} = 16$

Step 2

Replace all occurrences of $y$ with $1 + \frac{x}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} - 4 y^{2} = 16$ with $1 + \frac{x}{2}$ .

$x^{2} - 4 {(1 + \frac{x}{2})}^{2} = 16$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} - 4 {(1 + \frac{x}{2})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(1 + \frac{x}{2})}^{2}$ as $(1 + \frac{x}{2}) (1 + \frac{x}{2})$ .

$x^{2} - 4 ((1 + \frac{x}{2}) (1 + \frac{x}{2})) = 16$

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

Step 2.2.1.1.2

Expand $(1 + \frac{x}{2}) (1 + \frac{x}{2})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} - 4 (1 (1 + \frac{x}{2}) + \frac{x}{2} \cdot (1 + \frac{x}{2})) = 16$

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

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} - 4 (1 \cdot 1 + 1 (\frac{x}{2}) + \frac{x}{2} \cdot (1 + \frac{x}{2})) = 16$

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

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} - 4 (1 \cdot 1 + 1 (\frac{x}{2}) + \frac{x}{2} \cdot 1 + \frac{x}{2} \cdot \frac{x}{2}) = 16$

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

$x^{2} - 4 (1 \cdot 1 + 1 (\frac{x}{2}) + \frac{x}{2} \cdot 1 + \frac{x}{2} \cdot \frac{x}{2}) = 16$

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

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 $1$ by $1$ .

$x^{2} - 4 (1 + 1 (\frac{x}{2}) + \frac{x}{2} \cdot 1 + \frac{x}{2} \cdot \frac{x}{2}) = 16$

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

Step 2.2.1.1.3.1.2

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

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} \cdot 1 + \frac{x}{2} \cdot \frac{x}{2}) = 16$

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

Step 2.2.1.1.3.1.3

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

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x}{2} \cdot \frac{x}{2}) = 16$

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

Step 2.2.1.1.3.1.4

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

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Step 2.2.1.1.3.1.4.1

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

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x \cdot x}{2 \cdot 2}) = 16$

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

Step 2.2.1.1.3.1.4.2

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

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x \cdot x}{2 \cdot 2}) = 16$

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

Step 2.2.1.1.3.1.4.3

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

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x \cdot x}{2 \cdot 2}) = 16$

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

Step 2.2.1.1.3.1.4.4

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

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x^{1 + 1}}{2 \cdot 2}) = 16$

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

Step 2.2.1.1.3.1.4.5

Add $1$ and $1$ .

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x^{2}}{2 \cdot 2}) = 16$

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

Step 2.2.1.1.3.1.4.6

Multiply $2$ by $2$ .

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x^{2}}{4}) = 16$

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

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x^{2}}{4}) = 16$

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

$x^{2} - 4 (1 + \frac{x}{2} + \frac{x}{2} + \frac{x^{2}}{4}) = 16$

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

Step 2.2.1.1.3.2

Add $\frac{x}{2}$ and $\frac{x}{2}$ .

$x^{2} - 4 (1 + 2 (\frac{x}{2}) + \frac{x^{2}}{4}) = 16$

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

$x^{2} - 4 (1 + 2 (\frac{x}{2}) + \frac{x^{2}}{4}) = 16$

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

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Cancel the common factor.

$x^{2} - 4 (1 + 2 (\frac{x}{2}) + \frac{x^{2}}{4}) = 16$

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

Step 2.2.1.1.4.2

Rewrite the expression.

$x^{2} - 4 (1 + x + \frac{x^{2}}{4}) = 16$

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

$x^{2} - 4 (1 + x + \frac{x^{2}}{4}) = 16$

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

Step 2.2.1.1.5

Apply the distributive property.

$x^{2} - 4 \cdot 1 - 4 x - 4 \frac{x^{2}}{4} = 16$

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

Step 2.2.1.1.6

Simplify.

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Step 2.2.1.1.6.1

Multiply $- 4$ by $1$ .

$x^{2} - 4 - 4 x - 4 \frac{x^{2}}{4} = 16$

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

Step 2.2.1.1.6.2

Cancel the common factor of $4$ .

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Step 2.2.1.1.6.2.1

Factor $4$ out of $- 4$ .

$x^{2} - 4 - 4 x + 4 (- 1) (\frac{x^{2}}{4}) = 16$

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

Step 2.2.1.1.6.2.2

Cancel the common factor.

$x^{2} - 4 - 4 x + 4 \cdot (- 1 \frac{x^{2}}{4}) = 16$

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

Step 2.2.1.1.6.2.3

Rewrite the expression.

$x^{2} - 4 - 4 x - 1 x^{2} = 16$

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

$x^{2} - 4 - 4 x - 1 x^{2} = 16$

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

$x^{2} - 4 - 4 x - 1 x^{2} = 16$

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

Step 2.2.1.1.7

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

$x^{2} - 4 - 4 x - x^{2} = 16$

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

$x^{2} - 4 - 4 x - x^{2} = 16$

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

Step 2.2.1.2

Combine the opposite terms in $x^{2} - 4 - 4 x - x^{2}$ .

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Step 2.2.1.2.1

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

$- 4 - 4 x + 0 = 16$

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

Step 2.2.1.2.2

Add $- 4 - 4 x$ and $0$ .

$- 4 - 4 x = 16$

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

$- 4 - 4 x = 16$

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

$- 4 - 4 x = 16$

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

$- 4 - 4 x = 16$

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

$- 4 - 4 x = 16$

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

Step 3

Solve for $x$ in $- 4 - 4 x = 16$ .

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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

Add $4$ to both sides of the equation.

$- 4 x = 16 + 4$

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

Step 3.1.2

Add $16$ and $4$ .

$- 4 x = 20$

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

$- 4 x = 20$

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

Step 3.2

Divide each term in $- 4 x = 20$ by $- 4$ and simplify.

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Step 3.2.1

Divide each term in $- 4 x = 20$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{20}{- 4}$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{20}{- 4}$

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{20}{- 4}$

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

$x = \frac{20}{- 4}$

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

$x = \frac{20}{- 4}$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $20$ by $- 4$ .

$x = - 5$

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

$x = - 5$

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

$x = - 5$

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

$x = - 5$

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

Step 4

Replace all occurrences of $x$ with $- 5$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 1 + \frac{x}{2}$ with $- 5$ .

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

$x = - 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $1 + \frac{- 5}{2}$ .

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Step 4.2.1.1

Move the negative in front of the fraction.

$y = 1 - \frac{5}{2}$

$x = - 5$

Step 4.2.1.2

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

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

$x = - 5$

Step 4.2.1.3

Combine the numerators over the common denominator.

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

$x = - 5$

Step 4.2.1.4

Subtract $5$ from $2$ .

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

$x = - 5$

Step 4.2.1.5

Move the negative in front of the fraction.

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

$x = - 5$

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

$x = - 5$

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

$x = - 5$

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

$x = - 5$

Step 5

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

$(- 5, - \frac{3}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 5, - \frac{3}{2})$

Equation Form:

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

Step 7