Algebra Examples

$x^{2} + {(y + 2)}^{2} = 80$ $x - y = - 2$

Step 1

Add $y$ to both sides of the equation.

$x = - 2 + y$

$x^{2} + {(y + 2)}^{2} = 80$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + {(y + 2)}^{2} = 80$ with $- 2 + y$ .

${(- 2 + y)}^{2} + {(y + 2)}^{2} = 80$

$x = - 2 + y$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$(- 2 + y) (- 2 + y) + {(y + 2)}^{2} = 80$

$x = - 2 + y$

Step 2.2.1.1.2

Expand $(- 2 + y) (- 2 + y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$- 2 (- 2 + y) + y (- 2 + y) + {(y + 2)}^{2} = 80$

$x = - 2 + y$

Step 2.2.1.1.2.2

Apply the distributive property.

$- 2 \cdot - 2 - 2 y + y (- 2 + y) + {(y + 2)}^{2} = 80$

$x = - 2 + y$

Step 2.2.1.1.2.3

Apply the distributive property.

$- 2 \cdot - 2 - 2 y + y \cdot - 2 + y \cdot y + {(y + 2)}^{2} = 80$

$x = - 2 + y$

$- 2 \cdot - 2 - 2 y + y \cdot - 2 + y \cdot y + {(y + 2)}^{2} = 80$

$x = - 2 + y$

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 $- 2$ by $- 2$ .

$4 - 2 y + y \cdot - 2 + y \cdot y + {(y + 2)}^{2} = 80$

$x = - 2 + y$

Step 2.2.1.1.3.1.2

Move $- 2$ to the left of $y$ .

$4 - 2 y - 2 \cdot y + y \cdot y + {(y + 2)}^{2} = 80$

$x = - 2 + y$

Step 2.2.1.1.3.1.3

Multiply $y$ by $y$ .

$4 - 2 y - 2 y + y^{2} + {(y + 2)}^{2} = 80$

$x = - 2 + y$

$4 - 2 y - 2 y + y^{2} + {(y + 2)}^{2} = 80$

$x = - 2 + y$

Step 2.2.1.1.3.2

Subtract $2 y$ from $- 2 y$ .

$4 - 4 y + y^{2} + {(y + 2)}^{2} = 80$

$x = - 2 + y$

$4 - 4 y + y^{2} + {(y + 2)}^{2} = 80$

$x = - 2 + y$

Step 2.2.1.1.4

Rewrite ${(y + 2)}^{2}$ as $(y + 2) (y + 2)$ .

$4 - 4 y + y^{2} + (y + 2) (y + 2) = 80$

$x = - 2 + y$

Step 2.2.1.1.5

Expand $(y + 2) (y + 2)$ using the FOIL Method.

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Step 2.2.1.1.5.1

Apply the distributive property.

$4 - 4 y + y^{2} + y (y + 2) + 2 (y + 2) = 80$

$x = - 2 + y$

Step 2.2.1.1.5.2

Apply the distributive property.

$4 - 4 y + y^{2} + y \cdot y + y \cdot 2 + 2 (y + 2) = 80$

$x = - 2 + y$

Step 2.2.1.1.5.3

Apply the distributive property.

$4 - 4 y + y^{2} + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2 = 80$

$x = - 2 + y$

$4 - 4 y + y^{2} + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2 = 80$

$x = - 2 + y$

Step 2.2.1.1.6

Simplify and combine like terms.

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Step 2.2.1.1.6.1

Simplify each term.

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Step 2.2.1.1.6.1.1

Multiply $y$ by $y$ .

$4 - 4 y + y^{2} + y^{2} + y \cdot 2 + 2 y + 2 \cdot 2 = 80$

$x = - 2 + y$

Step 2.2.1.1.6.1.2

Move $2$ to the left of $y$ .

$4 - 4 y + y^{2} + y^{2} + 2 \cdot y + 2 y + 2 \cdot 2 = 80$

$x = - 2 + y$

Step 2.2.1.1.6.1.3

Multiply $2$ by $2$ .

$4 - 4 y + y^{2} + y^{2} + 2 y + 2 y + 4 = 80$

$x = - 2 + y$

$4 - 4 y + y^{2} + y^{2} + 2 y + 2 y + 4 = 80$

$x = - 2 + y$

Step 2.2.1.1.6.2

Add $2 y$ and $2 y$ .

$4 - 4 y + y^{2} + y^{2} + 4 y + 4 = 80$

$x = - 2 + y$

$4 - 4 y + y^{2} + y^{2} + 4 y + 4 = 80$

$x = - 2 + y$

$4 - 4 y + y^{2} + y^{2} + 4 y + 4 = 80$

$x = - 2 + y$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Combine the opposite terms in $4 - 4 y + y^{2} + y^{2} + 4 y + 4$ .

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Step 2.2.1.2.1.1

Add $- 4 y$ and $4 y$ .

$4 + y^{2} + y^{2} + 0 + 4 = 80$

$x = - 2 + y$

Step 2.2.1.2.1.2

Add $4 + y^{2} + y^{2}$ and $0$ .

$4 + y^{2} + y^{2} + 4 = 80$

$x = - 2 + y$

$4 + y^{2} + y^{2} + 4 = 80$

$x = - 2 + y$

Step 2.2.1.2.2

Add $4$ and $4$ .

$y^{2} + y^{2} + 8 = 80$

$x = - 2 + y$

Step 2.2.1.2.3

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

$2 y^{2} + 8 = 80$

$x = - 2 + y$

$2 y^{2} + 8 = 80$

$x = - 2 + y$

$2 y^{2} + 8 = 80$

$x = - 2 + y$

$2 y^{2} + 8 = 80$

$x = - 2 + y$

$2 y^{2} + 8 = 80$

$x = - 2 + y$

Step 3

Solve for $y$ in $2 y^{2} + 8 = 80$ .

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Step 3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.1.1

Subtract $8$ from both sides of the equation.

$2 y^{2} = 80 - 8$

$x = - 2 + y$

Step 3.1.2

Subtract $8$ from $80$ .

$2 y^{2} = 72$

$x = - 2 + y$

$2 y^{2} = 72$

$x = - 2 + y$

Step 3.2

Divide each term in $2 y^{2} = 72$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 y^{2} = 72$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{72}{2}$

$x = - 2 + y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{72}{2}$

$x = - 2 + y$

Step 3.2.2.1.2

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

$y^{2} = \frac{72}{2}$

$x = - 2 + y$

$y^{2} = \frac{72}{2}$

$x = - 2 + y$

$y^{2} = \frac{72}{2}$

$x = - 2 + y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $72$ by $2$ .

$y^{2} = 36$

$x = - 2 + y$

$y^{2} = 36$

$x = - 2 + y$

$y^{2} = 36$

$x = - 2 + y$

Step 3.3

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

$y = \pm \sqrt{36}$

$x = - 2 + y$

Step 3.4

Simplify $\pm \sqrt{36}$ .

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Step 3.4.1

Rewrite $36$ as $6^{2}$ .

$y = \pm \sqrt{6^{2}}$

$x = - 2 + y$

Step 3.4.2

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

$y = \pm 6$

$x = - 2 + y$

$y = \pm 6$

$x = - 2 + y$

Step 3.5

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

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Step 3.5.1

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

$y = 6$

$x = - 2 + y$

Step 3.5.2

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

$y = - 6$

$x = - 2 + y$

Step 3.5.3

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

$y = 6, - 6$

$x = - 2 + y$

$y = 6, - 6$

$x = - 2 + y$

$y = 6, - 6$

$x = - 2 + y$

Step 4

Replace all occurrences of $y$ with $6$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - 2 + y$ with $6$ .

$x = - 2 + 6$

$y = 6$

Step 4.2

Simplify $x = - 2 + 6$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$x = - 2 + 6$

$y = 6$

$x = - 2 + 6$

$y = 6$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Add $- 2$ and $6$ .

$x = 4$

$y = 6$

$x = 4$

$y = 6$

$x = 4$

$y = 6$

$x = 4$

$y = 6$

Step 5

Replace all occurrences of $y$ with $- 6$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = - 2 + y$ with $- 6$ .

$x = - 2 - 6$

$y = - 6$

Step 5.2

Simplify $x = - 2 - 6$ .

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Remove parentheses.

$x = - 2 - 6$

$y = - 6$

$x = - 2 - 6$

$y = - 6$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Subtract $6$ from $- 2$ .

$x = - 8$

$y = - 6$

$x = - 8$

$y = - 6$

$x = - 8$

$y = - 6$

$x = - 8$

$y = - 6$

Step 6

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

$(4, 6)$

$(- 8, - 6)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(4, 6), (- 8, - 6)$

Equation Form:

$x = 4, y = 6$

$x = - 8, y = - 6$

Step 8