Precalculus Examples

$x^{2} + y^{2} = 13$ , $x^{2} - y = 7$

Step 1

Solve for $y$ in $x^{2} - y = 7$ .

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Step 1.1

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

$- y = 7 - x^{2}$

$x^{2} + y^{2} = 13$

Step 1.2

Divide each term in $- y = 7 - x^{2}$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- y = 7 - x^{2}$ by $- 1$ .

$\frac{- y}{- 1} = \frac{7}{- 1} + \frac{- x^{2}}{- 1}$

$x^{2} + y^{2} = 13$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{7}{- 1} + \frac{- x^{2}}{- 1}$

$x^{2} + y^{2} = 13$

Step 1.2.2.2

Divide $y$ by $1$ .

$y = \frac{7}{- 1} + \frac{- x^{2}}{- 1}$

$x^{2} + y^{2} = 13$

$y = \frac{7}{- 1} + \frac{- x^{2}}{- 1}$

$x^{2} + y^{2} = 13$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $7$ by $- 1$ .

$y = - 7 + \frac{- x^{2}}{- 1}$

$x^{2} + y^{2} = 13$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y = - 7 + \frac{x^{2}}{1}$

$x^{2} + y^{2} = 13$

Step 1.2.3.1.3

Divide $x^{2}$ by $1$ .

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

$x^{2} + y^{2} = 13$

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

$x^{2} + y^{2} = 13$

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

$x^{2} + y^{2} = 13$

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

$x^{2} + y^{2} = 13$

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

$x^{2} + y^{2} = 13$

Step 2

Replace all occurrences of $y$ with $- 7 + x^{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 13$ with $- 7 + x^{2}$ .

$x^{2} + {(- 7 + x^{2})}^{2} = 13$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} + {(- 7 + x^{2})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(- 7 + x^{2})}^{2}$ as $(- 7 + x^{2}) (- 7 + x^{2})$ .

$x^{2} + (- 7 + x^{2}) (- 7 + x^{2}) = 13$

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

Step 2.2.1.1.2

Expand $(- 7 + x^{2}) (- 7 + x^{2})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} - 7 (- 7 + x^{2}) + x^{2} (- 7 + x^{2}) = 13$

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

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} - 7 \cdot - 7 - 7 x^{2} + x^{2} (- 7 + x^{2}) = 13$

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

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} - 7 \cdot - 7 - 7 x^{2} + x^{2} \cdot - 7 + x^{2} x^{2} = 13$

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

$x^{2} - 7 \cdot - 7 - 7 x^{2} + x^{2} \cdot - 7 + x^{2} x^{2} = 13$

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

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 $- 7$ by $- 7$ .

$x^{2} + 49 - 7 x^{2} + x^{2} \cdot - 7 + x^{2} x^{2} = 13$

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

Step 2.2.1.1.3.1.2

Move $- 7$ to the left of $x^{2}$ .

$x^{2} + 49 - 7 x^{2} - 7 \cdot x^{2} + x^{2} x^{2} = 13$

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

Step 2.2.1.1.3.1.3

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

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Step 2.2.1.1.3.1.3.1

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

$x^{2} + 49 - 7 x^{2} - 7 x^{2} + x^{2 + 2} = 13$

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

Step 2.2.1.1.3.1.3.2

Add $2$ and $2$ .

$x^{2} + 49 - 7 x^{2} - 7 x^{2} + x^{4} = 13$

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

$x^{2} + 49 - 7 x^{2} - 7 x^{2} + x^{4} = 13$

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

$x^{2} + 49 - 7 x^{2} - 7 x^{2} + x^{4} = 13$

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

Step 2.2.1.1.3.2

Subtract $7 x^{2}$ from $- 7 x^{2}$ .

$x^{2} + 49 - 14 x^{2} + x^{4} = 13$

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

$x^{2} + 49 - 14 x^{2} + x^{4} = 13$

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

$x^{2} + 49 - 14 x^{2} + x^{4} = 13$

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

Step 2.2.1.2

Subtract $14 x^{2}$ from $x^{2}$ .

$- 13 x^{2} + 49 + x^{4} = 13$

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

$- 13 x^{2} + 49 + x^{4} = 13$

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

$- 13 x^{2} + 49 + x^{4} = 13$

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

$- 13 x^{2} + 49 + x^{4} = 13$

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

Step 3

Solve for $x$ in $- 13 x^{2} + 49 + x^{4} = 13$ .

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Step 3.1

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

$u^{2} - 13 u + 49 = 13$

$u = x^{2}$

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

Step 3.2

Subtract $13$ from both sides of the equation.

$u^{2} - 13 u + 49 - 13 = 0$

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

Step 3.3

Subtract $13$ from $49$ .

$u^{2} - 13 u + 36 = 0$

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

Step 3.4

Factor $u^{2} - 13 u + 36$ using the AC method.

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Step 3.4.1

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 $36$ and whose sum is $- 13$ .

$- 9, - 4$

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

Step 3.4.2

Write the factored form using these integers.

$(u - 9) (u - 4) = 0$

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

$(u - 9) (u - 4) = 0$

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

Step 3.5

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

$u - 9 = 0$

$u - 4 = 0$

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

Step 3.6

Set $u - 9$ equal to $0$ and solve for $u$ .

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Step 3.6.1

Set $u - 9$ equal to $0$ .

$u - 9 = 0$

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

Step 3.6.2

Add $9$ to both sides of the equation.

$u = 9$

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

$u = 9$

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

Step 3.7

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 3.7.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

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

Step 3.7.2

Add $4$ to both sides of the equation.

$u = 4$

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

$u = 4$

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

Step 3.8

The final solution is all the values that make $(u - 9) (u - 4) = 0$ true.

$u = 9, 4$

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

Step 3.9

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

$x^{2} = 9$

$x^{2} = 4$

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

Step 3.10

Solve the first equation for $x$ .

$x^{2} = 9$

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

Step 3.11

Solve the equation for $x$ .

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Step 3.11.1

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

$x = \pm \sqrt{9}$

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

Step 3.11.2

Simplify $\pm \sqrt{9}$ .

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Step 3.11.2.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

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

Step 3.11.2.2

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

$x = \pm 3$

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

$x = \pm 3$

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

Step 3.11.3

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

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Step 3.11.3.1

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

$x = 3$

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

Step 3.11.3.2

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

$x = - 3$

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

Step 3.11.3.3

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

$x = 3, - 3$

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

$x = 3, - 3$

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

$x = 3, - 3$

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

Step 3.12

Solve the second equation for $x$ .

$x^{2} = 4$

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

Step 3.13

Solve the equation for $x$ .

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Step 3.13.1

Remove parentheses.

$x^{2} = 4$

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

Step 3.13.2

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

$x = \pm \sqrt{4}$

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

Step 3.13.3

Simplify $\pm \sqrt{4}$ .

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Step 3.13.3.1

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

$x = \pm \sqrt{2^{2}}$

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

Step 3.13.3.2

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

$x = \pm 2$

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

$x = \pm 2$

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

Step 3.13.4

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

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Step 3.13.4.1

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

$x = 2$

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

Step 3.13.4.2

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

$x = - 2$

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

Step 3.13.4.3

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

$x = 2, - 2$

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

$x = 2, - 2$

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

$x = 2, - 2$

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

Step 3.14

The solution to $x^{4} - 13 x^{2} + 49 = 13$ is $x = 3, - 3, 2, - 2$ .

$x = 3, - 3, 2, - 2$

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

$x = 3, - 3, 2, - 2$

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

Step 4

Replace all occurrences of $x$ with $3$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $3$ .

$y = - 7 + {(3)}^{2}$

$x = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 7 + {(3)}^{2}$ .

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Step 4.2.1.1

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

$y = - 7 + 9$

$x = 3$

Step 4.2.1.2

Add $- 7$ and $9$ .

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

Step 5

Replace all occurrences of $x$ with $- 3$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $- 3$ .

$y = - 7 + {(- 3)}^{2}$

$x = - 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 7 + {(- 3)}^{2}$ .

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Step 5.2.1.1

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

$y = - 7 + 9$

$x = - 3$

Step 5.2.1.2

Add $- 7$ and $9$ .

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

Step 6

Replace all occurrences of $x$ with $3$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $3$ .

$y = - 7 + {(3)}^{2}$

$x = 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 7 + {(3)}^{2}$ .

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Step 6.2.1.1

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

$y = - 7 + 9$

$x = 3$

Step 6.2.1.2

Add $- 7$ and $9$ .

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

Step 7

Replace all occurrences of $x$ with $- 3$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $- 3$ .

$y = - 7 + {(- 3)}^{2}$

$x = - 3$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- 7 + {(- 3)}^{2}$ .

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Step 7.2.1.1

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

$y = - 7 + 9$

$x = - 3$

Step 7.2.1.2

Add $- 7$ and $9$ .

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

Step 8

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

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Step 8.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $2$ .

$y = - 7 + {(2)}^{2}$

$x = 2$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $- 7 + {(2)}^{2}$ .

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Step 8.2.1.1

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

$y = - 7 + 4$

$x = 2$

Step 8.2.1.2

Add $- 7$ and $4$ .

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

Step 9

Replace all occurrences of $x$ with $3$ in each equation.

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Step 9.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $3$ .

$y = - 7 + {(3)}^{2}$

$x = 3$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $- 7 + {(3)}^{2}$ .

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Step 9.2.1.1

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

$y = - 7 + 9$

$x = 3$

Step 9.2.1.2

Add $- 7$ and $9$ .

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

Step 10

Replace all occurrences of $x$ with $- 3$ in each equation.

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Step 10.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $- 3$ .

$y = - 7 + {(- 3)}^{2}$

$x = - 3$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $- 7 + {(- 3)}^{2}$ .

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Step 10.2.1.1

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

$y = - 7 + 9$

$x = - 3$

Step 10.2.1.2

Add $- 7$ and $9$ .

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

Step 11

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

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Step 11.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $2$ .

$y = - 7 + {(2)}^{2}$

$x = 2$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $- 7 + {(2)}^{2}$ .

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Step 11.2.1.1

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

$y = - 7 + 4$

$x = 2$

Step 11.2.1.2

Add $- 7$ and $4$ .

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

Step 12

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

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Step 12.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $- 2$ .

$y = - 7 + {(- 2)}^{2}$

$x = - 2$

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $- 7 + {(- 2)}^{2}$ .

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Step 12.2.1.1

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

$y = - 7 + 4$

$x = - 2$

Step 12.2.1.2

Add $- 7$ and $4$ .

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

Step 13

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

$(3, 2)$

$(- 3, 2)$

$(2, - 3)$

$(- 2, - 3)$

Step 14

The result can be shown in multiple forms.

Point Form:

$(3, 2), (- 3, 2), (2, - 3), (- 2, - 3)$

Equation Form:

$x = 3, y = 2$

$x = - 3, y = 2$

$x = 2, y = - 3$

$x = - 2, y = - 3$

Step 15