Precalculus Examples

$2 x^{2} + y^{2} = 33$ , $x^{2} + y^{2} + 2 y = 19$

Step 1

Solve for $y$ in $2 x^{2} + y^{2} = 33$ .

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Step 1.1

Subtract $2 x^{2}$ from both sides of the equation.

$y^{2} = 33 - 2 x^{2}$

$x^{2} + y^{2} + 2 y = 19$

Step 1.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{33 - 2 x^{2}}$

$x^{2} + y^{2} + 2 y = 19$

Step 1.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.3.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{33 - 2 x^{2}}$

$x^{2} + y^{2} + 2 y = 19$

Step 1.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{33 - 2 x^{2}}$

$x^{2} + y^{2} + 2 y = 19$

Step 1.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{33 - 2 x^{2}}$

$y = - \sqrt{33 - 2 x^{2}}$

$x^{2} + y^{2} + 2 y = 19$

$y = \sqrt{33 - 2 x^{2}}$

$y = - \sqrt{33 - 2 x^{2}}$

$x^{2} + y^{2} + 2 y = 19$

$y = \sqrt{33 - 2 x^{2}}$

$y = - \sqrt{33 - 2 x^{2}}$

$x^{2} + y^{2} + 2 y = 19$

Step 2

Solve the system $y = \sqrt{33 - 2 x^{2}}, x^{2} + y^{2} + 2 y = 19$ .

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Step 2.1

Replace all occurrences of $y$ with $\sqrt{33 - 2 x^{2}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $y$ in $x^{2} + y^{2} + 2 y = 19$ with $\sqrt{33 - 2 x^{2}}$ .

$x^{2} + {(\sqrt{33 - 2 x^{2}})}^{2} + 2 (\sqrt{33 - 2 x^{2}}) = 19$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $x^{2} + {(\sqrt{33 - 2 x^{2}})}^{2} + 2 (\sqrt{33 - 2 x^{2}})$ .

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Step 2.1.2.1.1

Rewrite ${\sqrt{33 - 2 x^{2}}}^{2}$ as $33 - 2 x^{2}$ .

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Step 2.1.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33 - 2 x^{2}}$ as ${(33 - 2 x^{2})}^{\frac{1}{2}}$ .

$x^{2} + {({(33 - 2 x^{2})}^{\frac{1}{2}})}^{2} + 2 (\sqrt{33 - 2 x^{2}}) = 19$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.1.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} + {(33 - 2 x^{2})}^{\frac{1}{2} \cdot 2} + 2 (\sqrt{33 - 2 x^{2}}) = 19$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.1.2.1.1.3

Combine $\frac{1}{2}$ and $2$ .

$x^{2} + {(33 - 2 x^{2})}^{\frac{2}{2}} + 2 (\sqrt{33 - 2 x^{2}}) = 19$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.1.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.4.1

Cancel the common factor.

$x^{2} + {(33 - 2 x^{2})}^{\frac{2}{2}} + 2 (\sqrt{33 - 2 x^{2}}) = 19$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.1.2.1.1.4.2

Rewrite the expression.

$x^{2} + 33 - 2 x^{2} + 2 (\sqrt{33 - 2 x^{2}}) = 19$

$y = \sqrt{33 - 2 x^{2}}$

$x^{2} + 33 - 2 x^{2} + 2 (\sqrt{33 - 2 x^{2}}) = 19$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.1.2.1.1.5

Simplify.

$x^{2} + 33 - 2 x^{2} + 2 (\sqrt{33 - 2 x^{2}}) = 19$

$y = \sqrt{33 - 2 x^{2}}$

$x^{2} + 33 - 2 x^{2} + 2 \sqrt{33 - 2 x^{2}} = 19$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.1.2.1.2

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

$- x^{2} + 33 + 2 \sqrt{33 - 2 x^{2}} = 19$

$y = \sqrt{33 - 2 x^{2}}$

$- x^{2} + 33 + 2 \sqrt{33 - 2 x^{2}} = 19$

$y = \sqrt{33 - 2 x^{2}}$

$- x^{2} + 33 + 2 \sqrt{33 - 2 x^{2}} = 19$

$y = \sqrt{33 - 2 x^{2}}$

$- x^{2} + 33 + 2 \sqrt{33 - 2 x^{2}} = 19$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 4$

$y = \sqrt{33 - 2 x^{2}}$

Step 2.3

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

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Step 2.3.1

Replace all occurrences of $x$ in $y = \sqrt{33 - 2 x^{2}}$ with $4$ .

$y = \sqrt{33 - 2 {(4)}^{2}}$

$x = 4$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{33 - 2 {(4)}^{2}}$ .

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Step 2.3.2.1.1

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

$y = \sqrt{33 - 2 \cdot 16}$

$x = 4$

Step 2.3.2.1.2

Multiply $- 2$ by $16$ .

$y = \sqrt{33 - 32}$

$x = 4$

Step 2.3.2.1.3

Subtract $32$ from $33$ .

$y = \sqrt{1}$

$x = 4$

Step 2.3.2.1.4

Any root of $1$ is $1$ .

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

Step 3

Solve the system $y = - \sqrt{33 - 2 x^{2}}, x^{2} + y^{2} + 2 y = 19$ .

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Step 3.1

Replace all occurrences of $y$ with $- \sqrt{33 - 2 x^{2}}$ in each equation.

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Step 3.1.1

Replace all occurrences of $y$ in $x^{2} + y^{2} + 2 y = 19$ with $- \sqrt{33 - 2 x^{2}}$ .

$x^{2} + {(- \sqrt{33 - 2 x^{2}})}^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $x^{2} + {(- \sqrt{33 - 2 x^{2}})}^{2} + 2 (- \sqrt{33 - 2 x^{2}})$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{33 - 2 x^{2}}$ .

$x^{2} + {(- 1)}^{2} {\sqrt{33 - 2 x^{2}}}^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} + 1 {\sqrt{33 - 2 x^{2}}}^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{33 - 2 x^{2}}}^{2}$ by $1$ .

$x^{2} + {\sqrt{33 - 2 x^{2}}}^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{33 - 2 x^{2}}}^{2}$ as $33 - 2 x^{2}$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33 - 2 x^{2}}$ as ${(33 - 2 x^{2})}^{\frac{1}{2}}$ .

$x^{2} + {({(33 - 2 x^{2})}^{\frac{1}{2}})}^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} + {(33 - 2 x^{2})}^{\frac{1}{2} \cdot 2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$x^{2} + {(33 - 2 x^{2})}^{\frac{2}{2}} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

$x^{2} + {(33 - 2 x^{2})}^{\frac{2}{2}} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$x^{2} + 33 - 2 x^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

$x^{2} + 33 - 2 x^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.4.5

Simplify.

$x^{2} + 33 - 2 x^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

$x^{2} + 33 - 2 x^{2} + 2 (- \sqrt{33 - 2 x^{2}}) = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.1.5

Multiply $- 1$ by $2$ .

$x^{2} + 33 - 2 x^{2} - 2 \sqrt{33 - 2 x^{2}} = 19$

$y = - \sqrt{33 - 2 x^{2}}$

$x^{2} + 33 - 2 x^{2} - 2 \sqrt{33 - 2 x^{2}} = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.1.2.1.2

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

$- x^{2} + 33 - 2 \sqrt{33 - 2 x^{2}} = 19$

$y = - \sqrt{33 - 2 x^{2}}$

$- x^{2} + 33 - 2 \sqrt{33 - 2 x^{2}} = 19$

$y = - \sqrt{33 - 2 x^{2}}$

$- x^{2} + 33 - 2 \sqrt{33 - 2 x^{2}} = 19$

$y = - \sqrt{33 - 2 x^{2}}$

$- x^{2} + 33 - 2 \sqrt{33 - 2 x^{2}} = 19$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - 2, 2$

$y = - \sqrt{33 - 2 x^{2}}$

Step 3.3

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

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Step 3.3.1

Replace all occurrences of $x$ in $y = - \sqrt{33 - 2 x^{2}}$ with $- 2$ .

$y = - \sqrt{33 - 2 {(- 2)}^{2}}$

$x = - 2$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{33 - 2 {(- 2)}^{2}}$ .

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Step 3.3.2.1.1

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

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Step 3.3.2.1.1.1

Multiply $- 2$ by ${(- 2)}^{2}$ .

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Step 3.3.2.1.1.1.1

Raise $- 2$ to the power of $1$ .

$y = - \sqrt{33 + (- 2) {(- 2)}^{2}}$

$x = - 2$

Step 3.3.2.1.1.1.2

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

$y = - \sqrt{33 + {(- 2)}^{1 + 2}}$

$x = - 2$

$y = - \sqrt{33 + {(- 2)}^{1 + 2}}$

$x = - 2$

Step 3.3.2.1.1.2

Add $1$ and $2$ .

$y = - \sqrt{33 + {(- 2)}^{3}}$

$x = - 2$

$y = - \sqrt{33 + {(- 2)}^{3}}$

$x = - 2$

Step 3.3.2.1.2

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

$y = - \sqrt{33 - 8}$

$x = - 2$

Step 3.3.2.1.3

Subtract $8$ from $33$ .

$y = - \sqrt{25}$

$x = - 2$

Step 3.3.2.1.4

Rewrite $25$ as $5^{2}$ .

$y = - \sqrt{5^{2}}$

$x = - 2$

Step 3.3.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$y = - 1 \cdot 5$

$x = - 2$

Step 3.3.2.1.6

Multiply $- 1$ by $5$ .

$y = - 5$

$x = - 2$

$y = - 5$

$x = - 2$

$y = - 5$

$x = - 2$

$y = - 5$

$x = - 2$

Step 3.4

Replace all occurrences of $x$ with $2$ in each equation.

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Step 3.4.1

Replace all occurrences of $x$ in $y = - \sqrt{33 - 2 x^{2}}$ with $2$ .

$y = - \sqrt{33 - 2 {(2)}^{2}}$

$x = 2$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{33 - 2 {(2)}^{2}}$ .

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Step 3.4.2.1.1

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

$y = - \sqrt{33 - 2 \cdot 4}$

$x = 2$

Step 3.4.2.1.2

Multiply $- 2$ by $4$ .

$y = - \sqrt{33 - 8}$

$x = 2$

Step 3.4.2.1.3

Subtract $8$ from $33$ .

$y = - \sqrt{25}$

$x = 2$

Step 3.4.2.1.4

Rewrite $25$ as $5^{2}$ .

$y = - \sqrt{5^{2}}$

$x = 2$

Step 3.4.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$y = - 1 \cdot 5$

$x = 2$

Step 3.4.2.1.6

Multiply $- 1$ by $5$ .

$y = - 5$

$x = 2$

$y = - 5$

$x = 2$

$y = - 5$

$x = 2$

$y = - 5$

$x = 2$

$y = - 5$

$x = 2$

Step 4

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

$(4, 1)$

$(- 2, - 5)$

$(2, - 5)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(4, 1), (4, 1), (- 2, - 5), (2, - 5)$

Equation Form:

$x = 4, y = 1$

$x = - 2, y = - 5$

$x = 2, y = - 5$

Step 6