Finite Math Examples

x + x y = 7

$x + x y = 7$ ,

x - 2 y = - 5

$x - 2 y = - 5$

Step 1

Add

2 y

$2 y$ to both sides of the equation.

x = - 5 + 2 y

$x = - 5 + 2 y$

x + x y = 7

$x + x y = 7$

Step 2

Replace all occurrences of

x

$x$ with

- 5 + 2 y

$- 5 + 2 y$ in each equation.

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Step 2.1

Replace all occurrences of

x

$x$ in

x + x y = 7

$x + x y = 7$ with

- 5 + 2 y

$- 5 + 2 y$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify

(- 5 + 2 y) + (- 5 + 2 y) \cdot y

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

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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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 2.2.1.1.2

Multiply

y

$y$ by

y

$y$ by adding the exponents.

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Step 2.2.1.1.2.1

Move

y

$y$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 2.2.1.1.2.2

Multiply

y

$y$ by

y

$y$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 2.2.1.2

Subtract

5 y

$5 y$ from

2 y

$2 y$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3

Solve for

y

$y$ in

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

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

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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

$7$ from both sides of the equation.

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.1.2

Subtract

7

$7$ from

- 5

$- 5$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.2

Use the quadratic formula to find the solutions.

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.3

Substitute the values

a = 2

$a = 2$ ,

b = - 3

$b = - 3$ , and

c = - 12

$c = - 12$ into the quadratic formula and solve for

y

$y$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

$- 3$ to the power of

2

$2$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.4.1.2

Multiply

- 4 \cdot 2 \cdot - 12

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

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Step 3.4.1.2.1

Multiply

- 4

$- 4$ by

2

$2$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.4.1.2.2

Multiply

- 8

$- 8$ by

- 12

$- 12$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.4.1.3

Add

9

$9$ and

96

$96$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.4.2

Multiply

2

$2$ by

2

$2$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.5

Simplify the expression to solve for the

+

$+$ portion of the

\pm

$\pm$ .

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.5.1.2

Multiply

- 4 \cdot 2 \cdot - 12

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

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Step 3.5.1.2.1

Multiply

- 4

$- 4$ by

2

$2$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

Step 3.5.1.2.2

Multiply

- 8

$- 8$ by

- 12

$- 12$ .

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

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

x = - 5 + 2 y

$x = - 5 + 2 y$

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

$x = - 5 + 2 y$

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