Finite Math Examples

$x + x y = 7$ , $x - 2 y = - 5$

Step 1

Add $2 y$ to both sides of the equation.

$x = - 5 + 2 y$

$x + x y = 7$

Step 2

Replace all occurrences of $x$ with $- 5 + 2 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x + x y = 7$ with $- 5 + 2 y$ .

$(- 5 + 2 y) + (- 5 + 2 y) \cdot y = 7$

$x = - 5 + 2 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(- 5 + 2 y) + (- 5 + 2 y) \cdot y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$- 5 + 2 y - 5 y + 2 y \cdot y = 7$

$x = - 5 + 2 y$

Step 2.2.1.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.1.2.1

Move $y$ .

$- 5 + 2 y - 5 y + 2 (y \cdot y) = 7$

$x = - 5 + 2 y$

Step 2.2.1.1.2.2

Multiply $y$ by $y$ .

$- 5 + 2 y - 5 y + 2 y^{2} = 7$

$x = - 5 + 2 y$

$- 5 + 2 y - 5 y + 2 y^{2} = 7$

$x = - 5 + 2 y$

$- 5 + 2 y - 5 y + 2 y^{2} = 7$

$x = - 5 + 2 y$

Step 2.2.1.2

Subtract $5 y$ from $2 y$ .

$- 5 - 3 y + 2 y^{2} = 7$

$x = - 5 + 2 y$

$- 5 - 3 y + 2 y^{2} = 7$

$x = - 5 + 2 y$

$- 5 - 3 y + 2 y^{2} = 7$

$x = - 5 + 2 y$

$- 5 - 3 y + 2 y^{2} = 7$

$x = - 5 + 2 y$

Step 3

Solve for $y$ in $- 5 - 3 y + 2 y^{2} = 7$ .

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Step 3.1

Move all terms to the left side of the equation and simplify.

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Step 3.1.1

Subtract $7$ from both sides of the equation.

$- 5 - 3 y + 2 y^{2} - 7 = 0$

$x = - 5 + 2 y$

Step 3.1.2

Subtract $7$ from $- 5$ .

$- 3 y + 2 y^{2} - 12 = 0$

$x = - 5 + 2 y$

$- 3 y + 2 y^{2} - 12 = 0$

$x = - 5 + 2 y$

Step 3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x = - 5 + 2 y$

Step 3.3

Substitute the values $a = 2$ , $b = - 3$ , and $c = - 12$ into the quadratic formula and solve for $y$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot - 12)}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $- 3$ to the power of $2$ .

$y = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 12}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.4.1.2

Multiply $- 4 \cdot 2 \cdot - 12$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{3 \pm \sqrt{9 - 8 \cdot - 12}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.4.1.2.2

Multiply $- 8$ by $- 12$ .

$y = \frac{3 \pm \sqrt{9 + 96}}{2 \cdot 2}$

$x = - 5 + 2 y$

$y = \frac{3 \pm \sqrt{9 + 96}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.4.1.3

Add $9$ and $96$ .

$y = \frac{3 \pm \sqrt{105}}{2 \cdot 2}$

$x = - 5 + 2 y$

$y = \frac{3 \pm \sqrt{105}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.4.2

Multiply $2$ by $2$ .

$y = \frac{3 \pm \sqrt{105}}{4}$

$x = - 5 + 2 y$

$y = \frac{3 \pm \sqrt{105}}{4}$

$x = - 5 + 2 y$

Step 3.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 3$ to the power of $2$ .

$y = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 12}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.5.1.2

Multiply $- 4 \cdot 2 \cdot - 12$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{3 \pm \sqrt{9 - 8 \cdot - 12}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.5.1.2.2

Multiply $- 8$ by $- 12$ .

$y = \frac{3 \pm \sqrt{9 + 96}}{2 \cdot 2}$

$x = - 5 + 2 y$

$y = \frac{3 \pm \sqrt{9 + 96}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.5.1.3

Add $9$ and $96$ .

$y = \frac{3 \pm \sqrt{105}}{2 \cdot 2}$

$x = - 5 + 2 y$

$y = \frac{3 \pm \sqrt{105}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.5.2

Multiply $2$ by $2$ .

$y = \frac{3 \pm \sqrt{105}}{4}$

$x = - 5 + 2 y$

Step 3.5.3

Change the $\pm$ to $+$ .

$y = \frac{3 + \sqrt{105}}{4}$

$x = - 5 + 2 y$

$y = \frac{3 + \sqrt{105}}{4}$

$x = - 5 + 2 y$

Step 3.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 3$ to the power of $2$ .

$y = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 12}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.6.1.2

Multiply $- 4 \cdot 2 \cdot - 12$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{3 \pm \sqrt{9 - 8 \cdot - 12}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.6.1.2.2

Multiply $- 8$ by $- 12$ .

$y = \frac{3 \pm \sqrt{9 + 96}}{2 \cdot 2}$

$x = - 5 + 2 y$

$y = \frac{3 \pm \sqrt{9 + 96}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.6.1.3

Add $9$ and $96$ .

$y = \frac{3 \pm \sqrt{105}}{2 \cdot 2}$

$x = - 5 + 2 y$

$y = \frac{3 \pm \sqrt{105}}{2 \cdot 2}$

$x = - 5 + 2 y$

Step 3.6.2

Multiply $2$ by $2$ .

$y = \frac{3 \pm \sqrt{105}}{4}$

$x = - 5 + 2 y$

Step 3.6.3

Change the $\pm$ to $-$ .

$y = \frac{3 - \sqrt{105}}{4}$

$x = - 5 + 2 y$

$y = \frac{3 - \sqrt{105}}{4}$

$x = - 5 + 2 y$

Step 3.7

The final answer is the combination of both solutions.

$y = \frac{3 + \sqrt{105}}{4}, \frac{3 - \sqrt{105}}{4}$

$x = - 5 + 2 y$

$y = \frac{3 + \sqrt{105}}{4}, \frac{3 - \sqrt{105}}{4}$

$x = - 5 + 2 y$

Step 4

Replace all occurrences of $y$ with $\frac{3 + \sqrt{105}}{4}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - 5 + 2 y$ with $\frac{3 + \sqrt{105}}{4}$ .

$x = - 5 + 2 (\frac{3 + \sqrt{105}}{4})$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 5 + 2 (\frac{3 + \sqrt{105}}{4})$ .

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Factor $2$ out of $4$ .

$x = - 5 + 2 (\frac{3 + \sqrt{105}}{2 (2)})$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.1.2

Cancel the common factor.

$x = - 5 + 2 (\frac{3 + \sqrt{105}}{2 \cdot 2})$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.1.3

Rewrite the expression.

$x = - 5 + \frac{3 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

$x = - 5 + \frac{3 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - 5 \cdot \frac{2}{2} + \frac{3 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.3

Combine fractions.

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Step 4.2.1.3.1

Combine $- 5$ and $\frac{2}{2}$ .

$x = \frac{- 5 \cdot 2}{2} + \frac{3 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.3.2

Combine the numerators over the common denominator.

$x = \frac{- 5 \cdot 2 + 3 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

$x = \frac{- 5 \cdot 2 + 3 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.4

Simplify the numerator.

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Step 4.2.1.4.1

Multiply $- 5$ by $2$ .

$x = \frac{- 10 + 3 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.4.2

Add $- 10$ and $3$ .

$x = \frac{- 7 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

$x = \frac{- 7 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.5

Simplify with factoring out.

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Step 4.2.1.5.1

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.5.2

Factor $- 1$ out of $\sqrt{105}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{105})}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.5.3

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{105})$ .

$x = \frac{- 1 (7 - \sqrt{105})}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 4.2.1.5.4

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

$x = - \frac{7 - \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

$x = - \frac{7 - \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

$x = - \frac{7 - \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

$x = - \frac{7 - \sqrt{105}}{2}$

$y = \frac{3 + \sqrt{105}}{4}$

Step 5

Replace all occurrences of $y$ with $\frac{3 - \sqrt{105}}{4}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = - 5 + 2 y$ with $\frac{3 - \sqrt{105}}{4}$ .

$x = - 5 + 2 (\frac{3 - \sqrt{105}}{4})$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 5 + 2 (\frac{3 - \sqrt{105}}{4})$ .

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Factor $2$ out of $4$ .

$x = - 5 + 2 (\frac{3 - \sqrt{105}}{2 (2)})$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.1.2

Cancel the common factor.

$x = - 5 + 2 (\frac{3 - \sqrt{105}}{2 \cdot 2})$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.1.3

Rewrite the expression.

$x = - 5 + \frac{3 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

$x = - 5 + \frac{3 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - 5 \cdot \frac{2}{2} + \frac{3 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.3

Combine fractions.

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Step 5.2.1.3.1

Combine $- 5$ and $\frac{2}{2}$ .

$x = \frac{- 5 \cdot 2}{2} + \frac{3 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.3.2

Combine the numerators over the common denominator.

$x = \frac{- 5 \cdot 2 + 3 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

$x = \frac{- 5 \cdot 2 + 3 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.4

Simplify the numerator.

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Step 5.2.1.4.1

Multiply $- 5$ by $2$ .

$x = \frac{- 10 + 3 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.4.2

Add $- 10$ and $3$ .

$x = \frac{- 7 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

$x = \frac{- 7 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.5

Simplify with factoring out.

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Step 5.2.1.5.1

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.5.2

Factor $- 1$ out of $- \sqrt{105}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{105})}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.5.3

Factor $- 1$ out of $- 1 (7) - (\sqrt{105})$ .

$x = \frac{- 1 (7 + \sqrt{105})}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 5.2.1.5.4

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

$x = - \frac{7 + \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

$x = - \frac{7 + \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

$x = - \frac{7 + \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

$x = - \frac{7 + \sqrt{105}}{2}$

$y = \frac{3 - \sqrt{105}}{4}$

Step 6

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

$(- \frac{7 - \sqrt{105}}{2}, \frac{3 + \sqrt{105}}{4})$

$(- \frac{7 + \sqrt{105}}{2}, \frac{3 - \sqrt{105}}{4})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- \frac{7 - \sqrt{105}}{2}, \frac{3 + \sqrt{105}}{4}), (- \frac{7 + \sqrt{105}}{2}, \frac{3 - \sqrt{105}}{4})$

Equation Form:

$x = - \frac{7 - \sqrt{105}}{2}, y = \frac{3 + \sqrt{105}}{4}$

$x = - \frac{7 + \sqrt{105}}{2}, y = \frac{3 - \sqrt{105}}{4}$

Step 8