Finite Math Examples

$x^{2} + y^{2} = 36$ , $x + y = 4$

Step 1

Subtract $y$ from both sides of the equation.

$x = 4 - y$

$x^{2} + y^{2} = 36$

Step 2

Replace all occurrences of $x$ with $4 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 36$ with $4 - y$ .

${(4 - y)}^{2} + y^{2} = 36$

$x = 4 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(4 - y)}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(4 - y)}^{2}$ as $(4 - y) (4 - y)$ .

$(4 - y) (4 - y) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.2

Expand $(4 - y) (4 - y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$4 (4 - y) - y (4 - y) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.2.2

Apply the distributive property.

$4 \cdot 4 + 4 (- y) - y (4 - y) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.2.3

Apply the distributive property.

$4 \cdot 4 + 4 (- y) - y \cdot 4 - y (- y) + y^{2} = 36$

$x = 4 - y$

$4 \cdot 4 + 4 (- y) - y \cdot 4 - y (- y) + y^{2} = 36$

$x = 4 - y$

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 $4$ by $4$ .

$16 + 4 (- y) - y \cdot 4 - y (- y) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.3.1.2

Multiply $- 1$ by $4$ .

$16 - 4 y - y \cdot 4 - y (- y) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.3.1.3

Multiply $4$ by $- 1$ .

$16 - 4 y - 4 y - y (- y) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$16 - 4 y - 4 y - 1 \cdot (- 1 y \cdot y) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.3.1.5

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

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Step 2.2.1.1.3.1.5.1

Move $y$ .

$16 - 4 y - 4 y - 1 \cdot (- 1 (y \cdot y)) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$16 - 4 y - 4 y - 1 \cdot (- 1 y^{2}) + y^{2} = 36$

$x = 4 - y$

$16 - 4 y - 4 y - 1 \cdot (- 1 y^{2}) + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$16 - 4 y - 4 y + 1 y^{2} + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.3.1.7

Multiply $y^{2}$ by $1$ .

$16 - 4 y - 4 y + y^{2} + y^{2} = 36$

$x = 4 - y$

$16 - 4 y - 4 y + y^{2} + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.1.3.2

Subtract $4 y$ from $- 4 y$ .

$16 - 8 y + y^{2} + y^{2} = 36$

$x = 4 - y$

$16 - 8 y + y^{2} + y^{2} = 36$

$x = 4 - y$

$16 - 8 y + y^{2} + y^{2} = 36$

$x = 4 - y$

Step 2.2.1.2

Add $y^{2}$ and $y^{2}$ .

$16 - 8 y + 2 y^{2} = 36$

$x = 4 - y$

$16 - 8 y + 2 y^{2} = 36$

$x = 4 - y$

$16 - 8 y + 2 y^{2} = 36$

$x = 4 - y$

$16 - 8 y + 2 y^{2} = 36$

$x = 4 - y$

Step 3

Solve for $y$ in $16 - 8 y + 2 y^{2} = 36$ .

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Step 3.1

Subtract $36$ from both sides of the equation.

$16 - 8 y + 2 y^{2} - 36 = 0$

$x = 4 - y$

Step 3.2

Subtract $36$ from $16$ .

$- 8 y + 2 y^{2} - 20 = 0$

$x = 4 - y$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $2$ out of $- 8 y + 2 y^{2} - 20$ .

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Step 3.3.1.1

Factor $2$ out of $- 8 y$ .

$2 (- 4 y) + 2 y^{2} - 20 = 0$

$x = 4 - y$

Step 3.3.1.2

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

$2 (- 4 y) + 2 (y^{2}) - 20 = 0$

$x = 4 - y$

Step 3.3.1.3

Factor $2$ out of $- 20$ .

$2 (- 4 y) + 2 y^{2} + 2 \cdot - 10 = 0$

$x = 4 - y$

Step 3.3.1.4

Factor $2$ out of $2 (- 4 y) + 2 y^{2}$ .

$2 (- 4 y + y^{2}) + 2 \cdot - 10 = 0$

$x = 4 - y$

Step 3.3.1.5

Factor $2$ out of $2 (- 4 y + y^{2}) + 2 \cdot - 10$ .

$2 (- 4 y + y^{2} - 10) = 0$

$x = 4 - y$

$2 (- 4 y + y^{2} - 10) = 0$

$x = 4 - y$

Step 3.3.2

Reorder terms.

$2 (y^{2} - 4 y - 10) = 0$

$x = 4 - y$

$2 (y^{2} - 4 y - 10) = 0$

$x = 4 - y$

Step 3.4

Divide each term in $2 (y^{2} - 4 y - 10) = 0$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 (y^{2} - 4 y - 10) = 0$ by $2$ .

$\frac{2 (y^{2} - 4 y - 10)}{2} = \frac{0}{2}$

$x = 4 - y$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 (y^{2} - 4 y - 10)}{2} = \frac{0}{2}$

$x = 4 - y$

Step 3.4.2.1.2

Divide $y^{2} - 4 y - 10$ by $1$ .

$y^{2} - 4 y - 10 = \frac{0}{2}$

$x = 4 - y$

$y^{2} - 4 y - 10 = \frac{0}{2}$

$x = 4 - y$

$y^{2} - 4 y - 10 = \frac{0}{2}$

$x = 4 - y$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $2$ .

$y^{2} - 4 y - 10 = 0$

$x = 4 - y$

$y^{2} - 4 y - 10 = 0$

$x = 4 - y$

$y^{2} - 4 y - 10 = 0$

$x = 4 - y$

Step 3.5

Use the quadratic formula to find the solutions.

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

$x = 4 - y$

Step 3.6

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 10$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 10)}}{2 \cdot 1}$

$x = 4 - y$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $- 4$ to the power of $2$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 10}}{2 \cdot 1}$

$x = 4 - y$

Step 3.7.1.2

Multiply $- 4 \cdot 1 \cdot - 10$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot - 10}}{2 \cdot 1}$

$x = 4 - y$

Step 3.7.1.2.2

Multiply $- 4$ by $- 10$ .

$y = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot 1}$

$x = 4 - y$

Step 3.7.1.3

Add $16$ and $40$ .

$y = \frac{4 \pm \sqrt{56}}{2 \cdot 1}$

$x = 4 - y$

Step 3.7.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 3.7.1.4.1

Factor $4$ out of $56$ .

$y = \frac{4 \pm \sqrt{4 (14)}}{2 \cdot 1}$

$x = 4 - y$

Step 3.7.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

$x = 4 - y$

Step 3.7.1.5

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{14}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm 2 \sqrt{14}}{2 \cdot 1}$

$x = 4 - y$

Step 3.7.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{14}}{2}$

$x = 4 - y$

Step 3.7.3

Simplify $\frac{4 \pm 2 \sqrt{14}}{2}$ .

$y = 2 \pm \sqrt{14}$

$x = 4 - y$

$y = 2 \pm \sqrt{14}$

$x = 4 - y$

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Raise $- 4$ to the power of $2$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 10}}{2 \cdot 1}$

$x = 4 - y$

Step 3.8.1.2

Multiply $- 4 \cdot 1 \cdot - 10$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot - 10}}{2 \cdot 1}$

$x = 4 - y$

Step 3.8.1.2.2

Multiply $- 4$ by $- 10$ .

$y = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot 1}$

$x = 4 - y$

Step 3.8.1.3

Add $16$ and $40$ .

$y = \frac{4 \pm \sqrt{56}}{2 \cdot 1}$

$x = 4 - y$

Step 3.8.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 3.8.1.4.1

Factor $4$ out of $56$ .

$y = \frac{4 \pm \sqrt{4 (14)}}{2 \cdot 1}$

$x = 4 - y$

Step 3.8.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

$x = 4 - y$

Step 3.8.1.5

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{14}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm 2 \sqrt{14}}{2 \cdot 1}$

$x = 4 - y$

Step 3.8.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{14}}{2}$

$x = 4 - y$

Step 3.8.3

Simplify $\frac{4 \pm 2 \sqrt{14}}{2}$ .

$y = 2 \pm \sqrt{14}$

$x = 4 - y$

Step 3.8.4

Change the $\pm$ to $+$ .

$y = 2 + \sqrt{14}$

$x = 4 - y$

$y = 2 + \sqrt{14}$

$x = 4 - y$

Step 3.9

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

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Step 3.9.1

Simplify the numerator.

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Step 3.9.1.1

Raise $- 4$ to the power of $2$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 10}}{2 \cdot 1}$

$x = 4 - y$

Step 3.9.1.2

Multiply $- 4 \cdot 1 \cdot - 10$ .

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Step 3.9.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot - 10}}{2 \cdot 1}$

$x = 4 - y$

Step 3.9.1.2.2

Multiply $- 4$ by $- 10$ .

$y = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot 1}$

$x = 4 - y$

Step 3.9.1.3

Add $16$ and $40$ .

$y = \frac{4 \pm \sqrt{56}}{2 \cdot 1}$

$x = 4 - y$

Step 3.9.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 3.9.1.4.1

Factor $4$ out of $56$ .

$y = \frac{4 \pm \sqrt{4 (14)}}{2 \cdot 1}$

$x = 4 - y$

Step 3.9.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

$x = 4 - y$

Step 3.9.1.5

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{14}}{2 \cdot 1}$

$x = 4 - y$

$y = \frac{4 \pm 2 \sqrt{14}}{2 \cdot 1}$

$x = 4 - y$

Step 3.9.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{14}}{2}$

$x = 4 - y$

Step 3.9.3

Simplify $\frac{4 \pm 2 \sqrt{14}}{2}$ .

$y = 2 \pm \sqrt{14}$

$x = 4 - y$

Step 3.9.4

Change the $\pm$ to $-$ .

$y = 2 - \sqrt{14}$

$x = 4 - y$

$y = 2 - \sqrt{14}$

$x = 4 - y$

Step 3.10

The final answer is the combination of both solutions.

$y = 2 + \sqrt{14}, 2 - \sqrt{14}$

$x = 4 - y$

$y = 2 + \sqrt{14}, 2 - \sqrt{14}$

$x = 4 - y$

Step 4

Replace all occurrences of $y$ with $2 + \sqrt{14}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 4 - y$ with $2 + \sqrt{14}$ .

$x = 4 - (2 + \sqrt{14})$

$y = 2 + \sqrt{14}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $4 - (2 + \sqrt{14})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$x = 4 - 1 \cdot 2 - \sqrt{14}$

$y = 2 + \sqrt{14}$

Step 4.2.1.1.2

Multiply $- 1$ by $2$ .

$x = 4 - 2 - \sqrt{14}$

$y = 2 + \sqrt{14}$

$x = 4 - 2 - \sqrt{14}$

$y = 2 + \sqrt{14}$

Step 4.2.1.2

Subtract $2$ from $4$ .

$x = 2 - \sqrt{14}$

$y = 2 + \sqrt{14}$

$x = 2 - \sqrt{14}$

$y = 2 + \sqrt{14}$

$x = 2 - \sqrt{14}$

$y = 2 + \sqrt{14}$

$x = 2 - \sqrt{14}$

$y = 2 + \sqrt{14}$

Step 5

Replace all occurrences of $y$ with $2 - \sqrt{14}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 4 - y$ with $2 - \sqrt{14}$ .

$x = 4 - (2 - \sqrt{14})$

$y = 2 - \sqrt{14}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $4 - (2 - \sqrt{14})$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$x = 4 - 1 \cdot 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

Step 5.2.1.1.2

Multiply $- 1$ by $2$ .

$x = 4 - 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

Step 5.2.1.1.3

Multiply $- - \sqrt{14}$ .

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Step 5.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$x = 4 - 2 + 1 \sqrt{14}$

$y = 2 - \sqrt{14}$

Step 5.2.1.1.3.2

Multiply $\sqrt{14}$ by $1$ .

$x = 4 - 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

$x = 4 - 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

$x = 4 - 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

Step 5.2.1.2

Subtract $2$ from $4$ .

$x = 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

$x = 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

$x = 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

$x = 2 + \sqrt{14}$

$y = 2 - \sqrt{14}$

Step 6

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

$(2 - \sqrt{14}, 2 + \sqrt{14})$

$(2 + \sqrt{14}, 2 - \sqrt{14})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(2 - \sqrt{14}, 2 + \sqrt{14}), (2 + \sqrt{14}, 2 - \sqrt{14})$

Equation Form:

$x = 2 - \sqrt{14}, y = 2 + \sqrt{14}$

$x = 2 + \sqrt{14}, y = 2 - \sqrt{14}$

Step 8