Precalculus Examples

$x y = 9$ , $x^{2} + y^{2} = 82$

Step 1

Divide each term in $x y = 9$ by $y$ and simplify.

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Step 1.1

Divide each term in $x y = 9$ by $y$ .

$\frac{x y}{y} = \frac{9}{y}$

$x^{2} + y^{2} = 82$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $y$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{x y}{y} = \frac{9}{y}$

$x^{2} + y^{2} = 82$

Step 1.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{y}$

$x^{2} + y^{2} = 82$

$x = \frac{9}{y}$

$x^{2} + y^{2} = 82$

$x = \frac{9}{y}$

$x^{2} + y^{2} = 82$

$x = \frac{9}{y}$

$x^{2} + y^{2} = 82$

Step 2

Replace all occurrences of $x$ with $\frac{9}{y}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 82$ with $\frac{9}{y}$ .

${(\frac{9}{y})}^{2} + y^{2} = 82$

$x = \frac{9}{y}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the product rule to $\frac{9}{y}$ .

$\frac{9^{2}}{y^{2}} + y^{2} = 82$

$x = \frac{9}{y}$

Step 2.2.1.2

Raise $9$ to the power of $2$ .

$\frac{81}{y^{2}} + y^{2} = 82$

$x = \frac{9}{y}$

$\frac{81}{y^{2}} + y^{2} = 82$

$x = \frac{9}{y}$

$\frac{81}{y^{2}} + y^{2} = 82$

$x = \frac{9}{y}$

$\frac{81}{y^{2}} + y^{2} = 82$

$x = \frac{9}{y}$

Step 3

Solve for $y$ in $\frac{81}{y^{2}} + y^{2} = 82$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{2}, 1, 1$

$x = \frac{9}{y}$

Step 3.1.2

The LCM of one and any expression is the expression.

$y^{2}$

$x = \frac{9}{y}$

$y^{2}$

$x = \frac{9}{y}$

Step 3.2

Multiply each term in $\frac{81}{y^{2}} + y^{2} = 82$ by $y^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{81}{y^{2}} + y^{2} = 82$ by $y^{2}$ .

$\frac{81}{y^{2}} \cdot y^{2} + y^{2} y^{2} = 82 y^{2}$

$x = \frac{9}{y}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $y^{2}$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{81}{y^{2}} \cdot y^{2} + y^{2} y^{2} = 82 y^{2}$

$x = \frac{9}{y}$

Step 3.2.2.1.1.2

Rewrite the expression.

$81 + y^{2} y^{2} = 82 y^{2}$

$x = \frac{9}{y}$

$81 + y^{2} y^{2} = 82 y^{2}$

$x = \frac{9}{y}$

Step 3.2.2.1.2

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 3.2.2.1.2.1

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

$81 + y^{2 + 2} = 82 y^{2}$

$x = \frac{9}{y}$

Step 3.2.2.1.2.2

Add $2$ and $2$ .

$81 + y^{4} = 82 y^{2}$

$x = \frac{9}{y}$

$81 + y^{4} = 82 y^{2}$

$x = \frac{9}{y}$

$81 + y^{4} = 82 y^{2}$

$x = \frac{9}{y}$

$81 + y^{4} = 82 y^{2}$

$x = \frac{9}{y}$

$81 + y^{4} = 82 y^{2}$

$x = \frac{9}{y}$

Step 3.3

Solve the equation.

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Step 3.3.1

Subtract $82 y^{2}$ from both sides of the equation.

$81 + y^{4} - 82 y^{2} = 0$

$x = \frac{9}{y}$

Step 3.3.2

Substitute $u = y^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 82 u + 81 = 0$

$u = y^{2}$

$x = \frac{9}{y}$

Step 3.3.3

Factor $u^{2} - 82 u + 81$ using the AC method.

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Step 3.3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $81$ and whose sum is $- 82$ .

$- 81, - 1$

$x = \frac{9}{y}$

Step 3.3.3.2

Write the factored form using these integers.

$(u - 81) (u - 1) = 0$

$x = \frac{9}{y}$

$(u - 81) (u - 1) = 0$

$x = \frac{9}{y}$

Step 3.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 81 = 0$

$u - 1 = 0$

$x = \frac{9}{y}$

Step 3.3.5

Set $u - 81$ equal to $0$ and solve for $u$ .

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Step 3.3.5.1

Set $u - 81$ equal to $0$ .

$u - 81 = 0$

$x = \frac{9}{y}$

Step 3.3.5.2

Add $81$ to both sides of the equation.

$u = 81$

$x = \frac{9}{y}$

$u = 81$

$x = \frac{9}{y}$

Step 3.3.6

Set $u - 1$ equal to $0$ and solve for $u$ .

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Step 3.3.6.1

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

$x = \frac{9}{y}$

Step 3.3.6.2

Add $1$ to both sides of the equation.

$u = 1$

$x = \frac{9}{y}$

$u = 1$

$x = \frac{9}{y}$

Step 3.3.7

The final solution is all the values that make $(u - 81) (u - 1) = 0$ true.

$u = 81, 1$

$x = \frac{9}{y}$

Step 3.3.8

Substitute the real value of $u = y^{2}$ back into the solved equation.

$y^{2} = 81$

$y^{2} = 1$

$x = \frac{9}{y}$

Step 3.3.9

Solve the first equation for $y$ .

$y^{2} = 81$

$x = \frac{9}{y}$

Step 3.3.10

Solve the equation for $y$ .

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Step 3.3.10.1

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

$y = \pm \sqrt{81}$

$x = \frac{9}{y}$

Step 3.3.10.2

Simplify $\pm \sqrt{81}$ .

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Step 3.3.10.2.1

Rewrite $81$ as $9^{2}$ .

$y = \pm \sqrt{9^{2}}$

$x = \frac{9}{y}$

Step 3.3.10.2.2

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

$y = \pm 9$

$x = \frac{9}{y}$

$y = \pm 9$

$x = \frac{9}{y}$

Step 3.3.10.3

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

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Step 3.3.10.3.1

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

$y = 9$

$x = \frac{9}{y}$

Step 3.3.10.3.2

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

$y = - 9$

$x = \frac{9}{y}$

Step 3.3.10.3.3

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

$y = 9, - 9$

$x = \frac{9}{y}$

$y = 9, - 9$

$x = \frac{9}{y}$

$y = 9, - 9$

$x = \frac{9}{y}$

Step 3.3.11

Solve the second equation for $y$ .

$y^{2} = 1$

$x = \frac{9}{y}$

Step 3.3.12

Solve the equation for $y$ .

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Step 3.3.12.1

Remove parentheses.

$y^{2} = 1$

$x = \frac{9}{y}$

Step 3.3.12.2

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

$y = \pm \sqrt{1}$

$x = \frac{9}{y}$

Step 3.3.12.3

Any root of $1$ is $1$ .

$y = \pm 1$

$x = \frac{9}{y}$

Step 3.3.12.4

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

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Step 3.3.12.4.1

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

$y = 1$

$x = \frac{9}{y}$

Step 3.3.12.4.2

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

$y = - 1$

$x = \frac{9}{y}$

Step 3.3.12.4.3

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

$y = 1, - 1$

$x = \frac{9}{y}$

$y = 1, - 1$

$x = \frac{9}{y}$

$y = 1, - 1$

$x = \frac{9}{y}$

Step 3.3.13

The solution to $y^{4} - 82 y^{2} + 81 = 0$ is $y = 9, - 9, 1, - 1$ .

$y = 9, - 9, 1, - 1$

$x = \frac{9}{y}$

$y = 9, - 9, 1, - 1$

$x = \frac{9}{y}$

$y = 9, - 9, 1, - 1$

$x = \frac{9}{y}$

Step 4

Replace all occurrences of $y$ with $9$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $9$ .

$x = \frac{9}{9}$

$y = 9$

Step 4.2

Simplify the right side.

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Step 4.2.1

Divide $9$ by $9$ .

$x = 1$

$y = 9$

$x = 1$

$y = 9$

$x = 1$

$y = 9$

Step 5

Replace all occurrences of $y$ with $- 9$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $- 9$ .

$x = \frac{9}{- 9}$

$y = - 9$

Step 5.2

Simplify the right side.

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Step 5.2.1

Divide $9$ by $- 9$ .

$x = - 1$

$y = - 9$

$x = - 1$

$y = - 9$

$x = - 1$

$y = - 9$

Step 6

Replace all occurrences of $y$ with $9$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $9$ .

$x = \frac{9}{9}$

$y = 9$

Step 6.2

Simplify the right side.

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Step 6.2.1

Divide $9$ by $9$ .

$x = 1$

$y = 9$

$x = 1$

$y = 9$

$x = 1$

$y = 9$

Step 7

Replace all occurrences of $y$ with $- 9$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $- 9$ .

$x = \frac{9}{- 9}$

$y = - 9$

Step 7.2

Simplify the right side.

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Step 7.2.1

Divide $9$ by $- 9$ .

$x = - 1$

$y = - 9$

$x = - 1$

$y = - 9$

$x = - 1$

$y = - 9$

Step 8

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

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Step 8.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $1$ .

$x = \frac{9}{1}$

$y = 1$

Step 8.2

Simplify the right side.

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Step 8.2.1

Divide $9$ by $1$ .

$x = 9$

$y = 1$

$x = 9$

$y = 1$

$x = 9$

$y = 1$

Step 9

Replace all occurrences of $y$ with $9$ in each equation.

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Step 9.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $9$ .

$x = \frac{9}{9}$

$y = 9$

Step 9.2

Simplify the right side.

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Step 9.2.1

Divide $9$ by $9$ .

$x = 1$

$y = 9$

$x = 1$

$y = 9$

$x = 1$

$y = 9$

Step 10

Replace all occurrences of $y$ with $- 9$ in each equation.

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Step 10.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $- 9$ .

$x = \frac{9}{- 9}$

$y = - 9$

Step 10.2

Simplify the right side.

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Step 10.2.1

Divide $9$ by $- 9$ .

$x = - 1$

$y = - 9$

$x = - 1$

$y = - 9$

$x = - 1$

$y = - 9$

Step 11

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

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Step 11.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $1$ .

$x = \frac{9}{1}$

$y = 1$

Step 11.2

Simplify the right side.

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Step 11.2.1

Divide $9$ by $1$ .

$x = 9$

$y = 1$

$x = 9$

$y = 1$

$x = 9$

$y = 1$

Step 12

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

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Step 12.1

Replace all occurrences of $y$ in $x = \frac{9}{y}$ with $- 1$ .

$x = \frac{9}{- 1}$

$y = - 1$

Step 12.2

Simplify the right side.

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Step 12.2.1

Divide $9$ by $- 1$ .

$x = - 9$

$y = - 1$

$x = - 9$

$y = - 1$

$x = - 9$

$y = - 1$

Step 13

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

$(1, 9)$

$(- 1, - 9)$

$(9, 1)$

$(- 9, - 1)$

Step 14

The result can be shown in multiple forms.

Point Form:

$(1, 9), (- 1, - 9), (9, 1), (- 9, - 1)$

Equation Form:

$x = 1, y = 9$

$x = - 1, y = - 9$

$x = 9, y = 1$

$x = - 9, y = - 1$

Step 15