Algebra Examples

$x^{2} + y^{2} = 169$ $x^{2} + y = 13$

Step 1

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

$y = 13 - x^{2}$

$x^{2} + y^{2} = 169$

Step 2

Replace all occurrences of $y$ with $13 - x^{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 169$ with $13 - x^{2}$ .

$x^{2} + {(13 - x^{2})}^{2} = 169$

$y = 13 - x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} + {(13 - x^{2})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(13 - x^{2})}^{2}$ as $(13 - x^{2}) (13 - x^{2})$ .

$x^{2} + (13 - x^{2}) (13 - x^{2}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.2

Expand $(13 - x^{2}) (13 - x^{2})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} + 13 (13 - x^{2}) - x^{2} (13 - x^{2}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} + 13 \cdot 13 + 13 (- x^{2}) - x^{2} (13 - x^{2}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} + 13 \cdot 13 + 13 (- x^{2}) - x^{2} \cdot 13 - x^{2} (- x^{2}) = 169$

$y = 13 - x^{2}$

$x^{2} + 13 \cdot 13 + 13 (- x^{2}) - x^{2} \cdot 13 - x^{2} (- x^{2}) = 169$

$y = 13 - x^{2}$

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 $13$ by $13$ .

$x^{2} + 169 + 13 (- x^{2}) - x^{2} \cdot 13 - x^{2} (- x^{2}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.1.2

Multiply $- 1$ by $13$ .

$x^{2} + 169 - 13 x^{2} - x^{2} \cdot 13 - x^{2} (- x^{2}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.1.3

Multiply $13$ by $- 1$ .

$x^{2} + 169 - 13 x^{2} - 13 x^{2} - x^{2} (- x^{2}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$x^{2} + 169 - 13 x^{2} - 13 x^{2} - 1 \cdot (- 1 x^{2} x^{2}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.1.5

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

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Step 2.2.1.1.3.1.5.1

Move $x^{2}$ .

$x^{2} + 169 - 13 x^{2} - 13 x^{2} - 1 \cdot (- 1 (x^{2} x^{2})) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.1.5.2

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

$x^{2} + 169 - 13 x^{2} - 13 x^{2} - 1 \cdot (- 1 x^{2 + 2}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.1.5.3

Add $2$ and $2$ .

$x^{2} + 169 - 13 x^{2} - 13 x^{2} - 1 \cdot (- 1 x^{4}) = 169$

$y = 13 - x^{2}$

$x^{2} + 169 - 13 x^{2} - 13 x^{2} - 1 \cdot (- 1 x^{4}) = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$x^{2} + 169 - 13 x^{2} - 13 x^{2} + 1 x^{4} = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.1.7

Multiply $x^{4}$ by $1$ .

$x^{2} + 169 - 13 x^{2} - 13 x^{2} + x^{4} = 169$

$y = 13 - x^{2}$

$x^{2} + 169 - 13 x^{2} - 13 x^{2} + x^{4} = 169$

$y = 13 - x^{2}$

Step 2.2.1.1.3.2

Subtract $13 x^{2}$ from $- 13 x^{2}$ .

$x^{2} + 169 - 26 x^{2} + x^{4} = 169$

$y = 13 - x^{2}$

$x^{2} + 169 - 26 x^{2} + x^{4} = 169$

$y = 13 - x^{2}$

$x^{2} + 169 - 26 x^{2} + x^{4} = 169$

$y = 13 - x^{2}$

Step 2.2.1.2

Subtract $26 x^{2}$ from $x^{2}$ .

$- 25 x^{2} + 169 + x^{4} = 169$

$y = 13 - x^{2}$

$- 25 x^{2} + 169 + x^{4} = 169$

$y = 13 - x^{2}$

$- 25 x^{2} + 169 + x^{4} = 169$

$y = 13 - x^{2}$

$- 25 x^{2} + 169 + x^{4} = 169$

$y = 13 - x^{2}$

Step 3

Solve for $x$ in $- 25 x^{2} + 169 + x^{4} = 169$ .

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Step 3.1

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

$u^{2} - 25 u + 169 = 169$

$u = x^{2}$

$y = 13 - x^{2}$

Step 3.2

Subtract $169$ from both sides of the equation.

$u^{2} - 25 u + 169 - 169 = 0$

$y = 13 - x^{2}$

Step 3.3

Combine the opposite terms in $u^{2} - 25 u + 169 - 169$ .

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Step 3.3.1

Subtract $169$ from $169$ .

$u^{2} - 25 u + 0 = 0$

$y = 13 - x^{2}$

Step 3.3.2

Add $u^{2} - 25 u$ and $0$ .

$u^{2} - 25 u = 0$

$y = 13 - x^{2}$

$u^{2} - 25 u = 0$

$y = 13 - x^{2}$

Step 3.4

Factor $u$ out of $u^{2} - 25 u$ .

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Step 3.4.1

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

$u \cdot u - 25 u = 0$

$y = 13 - x^{2}$

Step 3.4.2

Factor $u$ out of $- 25 u$ .

$u \cdot u + u \cdot - 25 = 0$

$y = 13 - x^{2}$

Step 3.4.3

Factor $u$ out of $u \cdot u + u \cdot - 25$ .

$u (u - 25) = 0$

$y = 13 - x^{2}$

$u (u - 25) = 0$

$y = 13 - x^{2}$

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 $0$ .

$u = 0$

$u - 25 = 0$

$y = 13 - x^{2}$

Step 3.6

Set $u$ equal to $0$ .

$u = 0$

$y = 13 - x^{2}$

Step 3.7

Set $u - 25$ equal to $0$ and solve for $u$ .

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Step 3.7.1

Set $u - 25$ equal to $0$ .

$u - 25 = 0$

$y = 13 - x^{2}$

Step 3.7.2

Add $25$ to both sides of the equation.

$u = 25$

$y = 13 - x^{2}$

$u = 25$

$y = 13 - x^{2}$

Step 3.8

The final solution is all the values that make $u (u - 25) = 0$ true.

$u = 0, 25$

$y = 13 - x^{2}$

Step 3.9

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

$x^{2} = 0$

$x^{2} = 25$

$y = 13 - x^{2}$

Step 3.10

Solve the first equation for $x$ .

$x^{2} = 0$

$y = 13 - x^{2}$

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.

$x = \pm \sqrt{0}$

$y = 13 - x^{2}$

Step 3.11.2

Simplify $\pm \sqrt{0}$ .

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Step 3.11.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

$y = 13 - x^{2}$

Step 3.11.2.2

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

$x = \pm 0$

$y = 13 - x^{2}$

Step 3.11.2.3

Plus or minus $0$ is $0$ .

$x = 0$

$y = 13 - x^{2}$

$x = 0$

$y = 13 - x^{2}$

$x = 0$

$y = 13 - x^{2}$

Step 3.12

Solve the second equation for $x$ .

$x^{2} = 25$

$y = 13 - x^{2}$

Step 3.13

Solve the equation for $x$ .

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Step 3.13.1

Remove parentheses.

$x^{2} = 25$

$y = 13 - x^{2}$

Step 3.13.2

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

$x = \pm \sqrt{25}$

$y = 13 - x^{2}$

Step 3.13.3

Simplify $\pm \sqrt{25}$ .

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Step 3.13.3.1

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

$x = \pm \sqrt{5^{2}}$

$y = 13 - x^{2}$

Step 3.13.3.2

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

$x = \pm 5$

$y = 13 - x^{2}$

$x = \pm 5$

$y = 13 - x^{2}$

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.

$x = 5$

$y = 13 - x^{2}$

Step 3.13.4.2

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

$x = - 5$

$y = 13 - x^{2}$

Step 3.13.4.3

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

$x = 5, - 5$

$y = 13 - x^{2}$

$x = 5, - 5$

$y = 13 - x^{2}$

$x = 5, - 5$

$y = 13 - x^{2}$

Step 3.14

The solution to $x^{4} - 25 x^{2} + 169 = 169$ is $x = 0, 5, - 5$ .

$x = 0, 5, - 5$

$y = 13 - x^{2}$

$x = 0, 5, - 5$

$y = 13 - x^{2}$

Step 4

Replace all occurrences of $x$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 13 - x^{2}$ with $0$ .

$y = 13 - {(0)}^{2}$

$x = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $13 - {(0)}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$y = 13 - 0$

$x = 0$

Step 4.2.1.1.2

Multiply $- 1$ by $0$ .

$y = 13 + 0$

$x = 0$

$y = 13 + 0$

$x = 0$

Step 4.2.1.2

Add $13$ and $0$ .

$y = 13$

$x = 0$

$y = 13$

$x = 0$

$y = 13$

$x = 0$

$y = 13$

$x = 0$

Step 5

Replace all occurrences of $x$ with $5$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = 13 - x^{2}$ with $5$ .

$y = 13 - {(5)}^{2}$

$x = 5$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $13 - {(5)}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Raise $5$ to the power of $2$ .

$y = 13 - 1 \cdot 25$

$x = 5$

Step 5.2.1.1.2

Multiply $- 1$ by $25$ .

$y = 13 - 25$

$x = 5$

$y = 13 - 25$

$x = 5$

Step 5.2.1.2

Subtract $25$ from $13$ .

$y = - 12$

$x = 5$

$y = - 12$

$x = 5$

$y = - 12$

$x = 5$

$y = - 12$

$x = 5$

Step 6

Replace all occurrences of $x$ with $0$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = 13 - x^{2}$ with $0$ .

$y = 13 - {(0)}^{2}$

$x = 0$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $13 - {(0)}^{2}$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$y = 13 - 0$

$x = 0$

Step 6.2.1.1.2

Multiply $- 1$ by $0$ .

$y = 13 + 0$

$x = 0$

$y = 13 + 0$

$x = 0$

Step 6.2.1.2

Add $13$ and $0$ .

$y = 13$

$x = 0$

$y = 13$

$x = 0$

$y = 13$

$x = 0$

$y = 13$

$x = 0$

Step 7

Replace all occurrences of $x$ with $5$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = 13 - x^{2}$ with $5$ .

$y = 13 - {(5)}^{2}$

$x = 5$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $13 - {(5)}^{2}$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Raise $5$ to the power of $2$ .

$y = 13 - 1 \cdot 25$

$x = 5$

Step 7.2.1.1.2

Multiply $- 1$ by $25$ .

$y = 13 - 25$

$x = 5$

$y = 13 - 25$

$x = 5$

Step 7.2.1.2

Subtract $25$ from $13$ .

$y = - 12$

$x = 5$

$y = - 12$

$x = 5$

$y = - 12$

$x = 5$

$y = - 12$

$x = 5$

Step 8

Replace all occurrences of $x$ with $- 5$ in each equation.

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Step 8.1

Replace all occurrences of $x$ in $y = 13 - x^{2}$ with $- 5$ .

$y = 13 - {(- 5)}^{2}$

$x = - 5$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $13 - {(- 5)}^{2}$ .

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

Raise $- 5$ to the power of $2$ .

$y = 13 - 1 \cdot 25$

$x = - 5$

Step 8.2.1.1.2

Multiply $- 1$ by $25$ .

$y = 13 - 25$

$x = - 5$

$y = 13 - 25$

$x = - 5$

Step 8.2.1.2

Subtract $25$ from $13$ .

$y = - 12$

$x = - 5$

$y = - 12$

$x = - 5$

$y = - 12$

$x = - 5$

$y = - 12$

$x = - 5$

Step 9

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

$(0, 13)$

$(5, - 12)$

$(- 5, - 12)$

Step 10

The result can be shown in multiple forms.

Point Form:

$(0, 13), (5, - 12), (- 5, - 12)$

Equation Form:

$x = 0, y = 13$

$x = 5, y = - 12$

$x = - 5, y = - 12$

Step 11