Algebra Examples

${(x - 5)}^{2} + {(y - 3)}^{2} = 90$ $x + y = 2$

Step 1

Subtract $y$ from both sides of the equation.

$x = 2 - y$

${(x - 5)}^{2} + {(y - 3)}^{2} = 90$

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 ${(x - 5)}^{2} + {(y - 3)}^{2} = 90$ with $2 - y$ .

${((2 - y) - 5)}^{2} + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${((2 - y) - 5)}^{2} + {(y - 3)}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Subtract $5$ from $2$ .

${(- y - 3)}^{2} + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.2

Rewrite ${(- y - 3)}^{2}$ as $(- y - 3) (- y - 3)$ .

$(- y - 3) (- y - 3) + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.3

Expand $(- y - 3) (- y - 3)$ using the FOIL Method.

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Step 2.2.1.1.3.1

Apply the distributive property.

$- y (- y - 3) - 3 (- y - 3) + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.3.2

Apply the distributive property.

$- y (- y) - y \cdot - 3 - 3 (- y - 3) + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.3.3

Apply the distributive property.

$- y (- y) - y \cdot - 3 - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

$- y (- y) - y \cdot - 3 - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4

Simplify and combine like terms.

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Step 2.2.1.1.4.1

Simplify each term.

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Step 2.2.1.1.4.1.1

Rewrite using the commutative property of multiplication.

$- 1 \cdot (- 1 y \cdot y) - y \cdot - 3 - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.1.4.1.2.1

Move $y$ .

$- 1 \cdot (- 1 (y \cdot y)) - y \cdot - 3 - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4.1.2.2

Multiply $y$ by $y$ .

$- 1 \cdot (- 1 y^{2}) - y \cdot - 3 - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

$- 1 \cdot (- 1 y^{2}) - y \cdot - 3 - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4.1.3

Multiply $- 1$ by $- 1$ .

$1 y^{2} - y \cdot - 3 - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4.1.4

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

$y^{2} - y \cdot - 3 - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4.1.5

Multiply $- 3$ by $- 1$ .

$y^{2} + 3 y - 3 (- y) - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4.1.6

Multiply $- 1$ by $- 3$ .

$y^{2} + 3 y + 3 y - 3 \cdot - 3 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4.1.7

Multiply $- 3$ by $- 3$ .

$y^{2} + 3 y + 3 y + 9 + {(y - 3)}^{2} = 90$

$x = 2 - y$

$y^{2} + 3 y + 3 y + 9 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.4.2

Add $3 y$ and $3 y$ .

$y^{2} + 6 y + 9 + {(y - 3)}^{2} = 90$

$x = 2 - y$

$y^{2} + 6 y + 9 + {(y - 3)}^{2} = 90$

$x = 2 - y$

Step 2.2.1.1.5

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$y^{2} + 6 y + 9 + (y - 3) (y - 3) = 90$

$x = 2 - y$

Step 2.2.1.1.6

Expand $(y - 3) (y - 3)$ using the FOIL Method.

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Step 2.2.1.1.6.1

Apply the distributive property.

$y^{2} + 6 y + 9 + y (y - 3) - 3 (y - 3) = 90$

$x = 2 - y$

Step 2.2.1.1.6.2

Apply the distributive property.

$y^{2} + 6 y + 9 + y \cdot y + y \cdot - 3 - 3 (y - 3) = 90$

$x = 2 - y$

Step 2.2.1.1.6.3

Apply the distributive property.

$y^{2} + 6 y + 9 + y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3 = 90$

$x = 2 - y$

$y^{2} + 6 y + 9 + y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3 = 90$

$x = 2 - y$

Step 2.2.1.1.7

Simplify and combine like terms.

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Step 2.2.1.1.7.1

Simplify each term.

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Step 2.2.1.1.7.1.1

Multiply $y$ by $y$ .

$y^{2} + 6 y + 9 + y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3 = 90$

$x = 2 - y$

Step 2.2.1.1.7.1.2

Move $- 3$ to the left of $y$ .

$y^{2} + 6 y + 9 + y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3 = 90$

$x = 2 - y$

Step 2.2.1.1.7.1.3

Multiply $- 3$ by $- 3$ .

$y^{2} + 6 y + 9 + y^{2} - 3 y - 3 y + 9 = 90$

$x = 2 - y$

$y^{2} + 6 y + 9 + y^{2} - 3 y - 3 y + 9 = 90$

$x = 2 - y$

Step 2.2.1.1.7.2

Subtract $3 y$ from $- 3 y$ .

$y^{2} + 6 y + 9 + y^{2} - 6 y + 9 = 90$

$x = 2 - y$

$y^{2} + 6 y + 9 + y^{2} - 6 y + 9 = 90$

$x = 2 - y$

$y^{2} + 6 y + 9 + y^{2} - 6 y + 9 = 90$

$x = 2 - y$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Combine the opposite terms in $y^{2} + 6 y + 9 + y^{2} - 6 y + 9$ .

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Step 2.2.1.2.1.1

Subtract $6 y$ from $6 y$ .

$y^{2} + 9 + y^{2} + 0 + 9 = 90$

$x = 2 - y$

Step 2.2.1.2.1.2

Add $y^{2} + 9 + y^{2}$ and $0$ .

$y^{2} + 9 + y^{2} + 9 = 90$

$x = 2 - y$

$y^{2} + 9 + y^{2} + 9 = 90$

$x = 2 - y$

Step 2.2.1.2.2

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

$2 y^{2} + 9 + 9 = 90$

$x = 2 - y$

Step 2.2.1.2.3

Add $9$ and $9$ .

$2 y^{2} + 18 = 90$

$x = 2 - y$

$2 y^{2} + 18 = 90$

$x = 2 - y$

$2 y^{2} + 18 = 90$

$x = 2 - y$

$2 y^{2} + 18 = 90$

$x = 2 - y$

$2 y^{2} + 18 = 90$

$x = 2 - y$

Step 3

Solve for $y$ in $2 y^{2} + 18 = 90$ .

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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 $18$ from both sides of the equation.

$2 y^{2} = 90 - 18$

$x = 2 - y$

Step 3.1.2

Subtract $18$ from $90$ .

$2 y^{2} = 72$

$x = 2 - y$

$2 y^{2} = 72$

$x = 2 - y$

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 the right side.

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Step 4.2.1

Simplify $2 - (6)$ .

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Step 4.2.1.1

Multiply $- 1$ by $6$ .

$x = 2 - 6$

$y = 6$

Step 4.2.1.2

Subtract $6$ from $2$ .

$x = - 4$

$y = 6$

$x = - 4$

$y = 6$

$x = - 4$

$y = 6$

$x = - 4$

$y = 6$

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 the right side.

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Step 5.2.1

Simplify $2 - (- 6)$ .

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Step 5.2.1.1

Multiply $- 1$ by $- 6$ .

$x = 2 + 6$

$y = - 6$

Step 5.2.1.2

Add $2$ and $6$ .

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

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