Precalculus Examples

$x^{2} + y^{2} = 36$ , $x + y = 1$

Step 1

Subtract $y$ from both sides of the equation.

$x = 1 - y$

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

Step 2

Replace all occurrences of $x$ with $1 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 36$ with $1 - y$ .

${(1 - y)}^{2} + y^{2} = 36$

$x = 1 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(1 - y)}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(1 - y)}^{2}$ as $(1 - y) (1 - y)$ .

$(1 - y) (1 - y) + y^{2} = 36$

$x = 1 - y$

Step 2.2.1.1.2

Expand $(1 - y) (1 - y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$1 (1 - y) - y (1 - y) + y^{2} = 36$

$x = 1 - y$

Step 2.2.1.1.2.2

Apply the distributive property.

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

$x = 1 - y$

Step 2.2.1.1.2.3

Apply the distributive property.

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

$x = 1 - y$

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

$x = 1 - y$

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 $1$ by $1$ .

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

$x = 1 - y$

Step 2.2.1.1.3.1.2

Multiply $- y$ by $1$ .

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

$x = 1 - y$

Step 2.2.1.1.3.1.3

Multiply $- 1$ by $1$ .

$1 - y - y - y (- y) + y^{2} = 36$

$x = 1 - y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

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

$x = 1 - y$

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 $y$ .

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

$x = 1 - y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

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

$x = 1 - y$

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

$x = 1 - y$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

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

$x = 1 - y$

Step 2.2.1.1.3.1.7

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

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

$x = 1 - y$

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

$x = 1 - y$

Step 2.2.1.1.3.2

Subtract $y$ from $- y$ .

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

$x = 1 - y$

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

$x = 1 - y$

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

$x = 1 - y$

Step 2.2.1.2

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

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

$x = 1 - y$

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

$x = 1 - y$

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

$x = 1 - y$

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

$x = 1 - y$

Step 3

Solve for $y$ in $1 - 2 y + 2 y^{2} = 36$ .

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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 $36$ from both sides of the equation.

$1 - 2 y + 2 y^{2} - 36 = 0$

$x = 1 - y$

Step 3.1.2

Subtract $36$ from $1$ .

$- 2 y + 2 y^{2} - 35 = 0$

$x = 1 - y$

$- 2 y + 2 y^{2} - 35 = 0$

$x = 1 - y$

Step 3.2

Use the quadratic formula to find the solutions.

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

$x = 1 - y$

Step 3.3

Substitute the values $a = 2$ , $b = - 2$ , and $c = - 35$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot - 35)}}{2 \cdot 2}$

$x = 1 - y$

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 35}}{2 \cdot 2}$

$x = 1 - y$

Step 3.4.1.2

Multiply $- 4 \cdot 2 \cdot - 35$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{2 \pm \sqrt{4 - 8 \cdot - 35}}{2 \cdot 2}$

$x = 1 - y$

Step 3.4.1.2.2

Multiply $- 8$ by $- 35$ .

$y = \frac{2 \pm \sqrt{4 + 280}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm \sqrt{4 + 280}}{2 \cdot 2}$

$x = 1 - y$

Step 3.4.1.3

Add $4$ and $280$ .

$y = \frac{2 \pm \sqrt{284}}{2 \cdot 2}$

$x = 1 - y$

Step 3.4.1.4

Rewrite $284$ as $2^{2} \cdot 71$ .

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Step 3.4.1.4.1

Factor $4$ out of $284$ .

$y = \frac{2 \pm \sqrt{4 (71)}}{2 \cdot 2}$

$x = 1 - y$

Step 3.4.1.4.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot 71}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm \sqrt{2^{2} \cdot 71}}{2 \cdot 2}$

$x = 1 - y$

Step 3.4.1.5

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{71}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm 2 \sqrt{71}}{2 \cdot 2}$

$x = 1 - y$

Step 3.4.2

Multiply $2$ by $2$ .

$y = \frac{2 \pm 2 \sqrt{71}}{4}$

$x = 1 - y$

Step 3.4.3

Simplify $\frac{2 \pm 2 \sqrt{71}}{4}$ .

$y = \frac{1 \pm \sqrt{71}}{2}$

$x = 1 - y$

$y = \frac{1 \pm \sqrt{71}}{2}$

$x = 1 - y$

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 35}}{2 \cdot 2}$

$x = 1 - y$

Step 3.5.1.2

Multiply $- 4 \cdot 2 \cdot - 35$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{2 \pm \sqrt{4 - 8 \cdot - 35}}{2 \cdot 2}$

$x = 1 - y$

Step 3.5.1.2.2

Multiply $- 8$ by $- 35$ .

$y = \frac{2 \pm \sqrt{4 + 280}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm \sqrt{4 + 280}}{2 \cdot 2}$

$x = 1 - y$

Step 3.5.1.3

Add $4$ and $280$ .

$y = \frac{2 \pm \sqrt{284}}{2 \cdot 2}$

$x = 1 - y$

Step 3.5.1.4

Rewrite $284$ as $2^{2} \cdot 71$ .

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Step 3.5.1.4.1

Factor $4$ out of $284$ .

$y = \frac{2 \pm \sqrt{4 (71)}}{2 \cdot 2}$

$x = 1 - y$

Step 3.5.1.4.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot 71}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm \sqrt{2^{2} \cdot 71}}{2 \cdot 2}$

$x = 1 - y$

Step 3.5.1.5

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{71}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm 2 \sqrt{71}}{2 \cdot 2}$

$x = 1 - y$

Step 3.5.2

Multiply $2$ by $2$ .

$y = \frac{2 \pm 2 \sqrt{71}}{4}$

$x = 1 - y$

Step 3.5.3

Simplify $\frac{2 \pm 2 \sqrt{71}}{4}$ .

$y = \frac{1 \pm \sqrt{71}}{2}$

$x = 1 - y$

Step 3.5.4

Change the $\pm$ to $+$ .

$y = \frac{1 + \sqrt{71}}{2}$

$x = 1 - y$

$y = \frac{1 + \sqrt{71}}{2}$

$x = 1 - y$

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 35}}{2 \cdot 2}$

$x = 1 - y$

Step 3.6.1.2

Multiply $- 4 \cdot 2 \cdot - 35$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{2 \pm \sqrt{4 - 8 \cdot - 35}}{2 \cdot 2}$

$x = 1 - y$

Step 3.6.1.2.2

Multiply $- 8$ by $- 35$ .

$y = \frac{2 \pm \sqrt{4 + 280}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm \sqrt{4 + 280}}{2 \cdot 2}$

$x = 1 - y$

Step 3.6.1.3

Add $4$ and $280$ .

$y = \frac{2 \pm \sqrt{284}}{2 \cdot 2}$

$x = 1 - y$

Step 3.6.1.4

Rewrite $284$ as $2^{2} \cdot 71$ .

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Step 3.6.1.4.1

Factor $4$ out of $284$ .

$y = \frac{2 \pm \sqrt{4 (71)}}{2 \cdot 2}$

$x = 1 - y$

Step 3.6.1.4.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot 71}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm \sqrt{2^{2} \cdot 71}}{2 \cdot 2}$

$x = 1 - y$

Step 3.6.1.5

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{71}}{2 \cdot 2}$

$x = 1 - y$

$y = \frac{2 \pm 2 \sqrt{71}}{2 \cdot 2}$

$x = 1 - y$

Step 3.6.2

Multiply $2$ by $2$ .

$y = \frac{2 \pm 2 \sqrt{71}}{4}$

$x = 1 - y$

Step 3.6.3

Simplify $\frac{2 \pm 2 \sqrt{71}}{4}$ .

$y = \frac{1 \pm \sqrt{71}}{2}$

$x = 1 - y$

Step 3.6.4

Change the $\pm$ to $-$ .

$y = \frac{1 - \sqrt{71}}{2}$

$x = 1 - y$

$y = \frac{1 - \sqrt{71}}{2}$

$x = 1 - y$

Step 3.7

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{71}}{2}, \frac{1 - \sqrt{71}}{2}$

$x = 1 - y$

$y = \frac{1 + \sqrt{71}}{2}, \frac{1 - \sqrt{71}}{2}$

$x = 1 - y$

Step 4

Replace all occurrences of $y$ with $\frac{1 + \sqrt{71}}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 1 - y$ with $\frac{1 + \sqrt{71}}{2}$ .

$x = 1 - (\frac{1 + \sqrt{71}}{2})$

$y = \frac{1 + \sqrt{71}}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $1 - (\frac{1 + \sqrt{71}}{2})$ .

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Step 4.2.1.1

Simplify the expression.

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Step 4.2.1.1.1

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

$x = \frac{2}{2} - \frac{1 + \sqrt{71}}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

Step 4.2.1.1.2

Combine the numerators over the common denominator.

$x = \frac{2 - (1 + \sqrt{71})}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

$x = \frac{2 - (1 + \sqrt{71})}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

Step 4.2.1.2

Simplify the numerator.

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Step 4.2.1.2.1

Apply the distributive property.

$x = \frac{2 - 1 \cdot 1 - \sqrt{71}}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

Step 4.2.1.2.2

Multiply $- 1$ by $1$ .

$x = \frac{2 - 1 - \sqrt{71}}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

Step 4.2.1.2.3

Subtract $1$ from $2$ .

$x = \frac{1 - \sqrt{71}}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

$x = \frac{1 - \sqrt{71}}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

$x = \frac{1 - \sqrt{71}}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

$x = \frac{1 - \sqrt{71}}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

$x = \frac{1 - \sqrt{71}}{2}$

$y = \frac{1 + \sqrt{71}}{2}$

Step 5

Replace all occurrences of $y$ with $\frac{1 - \sqrt{71}}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 1 - y$ with $\frac{1 - \sqrt{71}}{2}$ .

$x = 1 - (\frac{1 - \sqrt{71}}{2})$

$y = \frac{1 - \sqrt{71}}{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $1 - (\frac{1 - \sqrt{71}}{2})$ .

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Step 5.2.1.1

Simplify the expression.

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Step 5.2.1.1.1

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

$x = \frac{2}{2} - \frac{1 - \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

Step 5.2.1.1.2

Combine the numerators over the common denominator.

$x = \frac{2 - (1 - \sqrt{71})}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

$x = \frac{2 - (1 - \sqrt{71})}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

Step 5.2.1.2

Simplify the numerator.

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Step 5.2.1.2.1

Apply the distributive property.

$x = \frac{2 - 1 \cdot 1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

Step 5.2.1.2.2

Multiply $- 1$ by $1$ .

$x = \frac{2 - 1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

Step 5.2.1.2.3

Multiply $- - \sqrt{71}$ .

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Step 5.2.1.2.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 - 1 + 1 \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

Step 5.2.1.2.3.2

Multiply $\sqrt{71}$ by $1$ .

$x = \frac{2 - 1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

$x = \frac{2 - 1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

Step 5.2.1.2.4

Subtract $1$ from $2$ .

$x = \frac{1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

$x = \frac{1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

$x = \frac{1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

$x = \frac{1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

$x = \frac{1 + \sqrt{71}}{2}$

$y = \frac{1 - \sqrt{71}}{2}$

Step 6

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

$(\frac{1 - \sqrt{71}}{2}, \frac{1 + \sqrt{71}}{2})$

$(\frac{1 + \sqrt{71}}{2}, \frac{1 - \sqrt{71}}{2})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{1 - \sqrt{71}}{2}, \frac{1 + \sqrt{71}}{2}), (\frac{1 + \sqrt{71}}{2}, \frac{1 - \sqrt{71}}{2})$

Equation Form:

$x = \frac{1 - \sqrt{71}}{2}, y = \frac{1 + \sqrt{71}}{2}$

$x = \frac{1 + \sqrt{71}}{2}, y = \frac{1 - \sqrt{71}}{2}$

Step 8