Algebra Examples

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

Step 1

Subtract $3 y$ from both sides of the equation.

$x = 5 - 3 y$

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

Step 2

Replace all occurrences of $x$ with $5 - 3 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in ${(x - 5)}^{2} + y^{2} = 90$ with $5 - 3 y$ .

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

$x = 5 - 3 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Combine the opposite terms in ${((5 - 3 y) - 5)}^{2} + y^{2}$ .

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Step 2.2.1.1.1

Subtract $5$ from $5$ .

${(- 3 y + 0)}^{2} + y^{2} = 90$

$x = 5 - 3 y$

Step 2.2.1.1.2

Add $- 3 y$ and $0$ .

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

$x = 5 - 3 y$

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

$x = 5 - 3 y$

Step 2.2.1.2

Simplify each term.

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Step 2.2.1.2.1

Apply the product rule to $- 3 y$ .

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

$x = 5 - 3 y$

Step 2.2.1.2.2

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

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

$x = 5 - 3 y$

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

$x = 5 - 3 y$

Step 2.2.1.3

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

$10 y^{2} = 90$

$x = 5 - 3 y$

$10 y^{2} = 90$

$x = 5 - 3 y$

$10 y^{2} = 90$

$x = 5 - 3 y$

$10 y^{2} = 90$

$x = 5 - 3 y$

Step 3

Solve for $y$ in $10 y^{2} = 90$ .

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Step 3.1

Divide each term in $10 y^{2} = 90$ by $10$ and simplify.

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Step 3.1.1

Divide each term in $10 y^{2} = 90$ by $10$ .

$\frac{10 y^{2}}{10} = \frac{90}{10}$

$x = 5 - 3 y$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $10$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{10 y^{2}}{10} = \frac{90}{10}$

$x = 5 - 3 y$

Step 3.1.2.1.2

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

$y^{2} = \frac{90}{10}$

$x = 5 - 3 y$

$y^{2} = \frac{90}{10}$

$x = 5 - 3 y$

$y^{2} = \frac{90}{10}$

$x = 5 - 3 y$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $90$ by $10$ .

$y^{2} = 9$

$x = 5 - 3 y$

$y^{2} = 9$

$x = 5 - 3 y$

$y^{2} = 9$

$x = 5 - 3 y$

Step 3.2

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

$y = \pm \sqrt{9}$

$x = 5 - 3 y$

Step 3.3

Simplify $\pm \sqrt{9}$ .

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Step 3.3.1

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

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

$x = 5 - 3 y$

Step 3.3.2

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

$y = \pm 3$

$x = 5 - 3 y$

$y = \pm 3$

$x = 5 - 3 y$

Step 3.4

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

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Step 3.4.1

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

$y = 3$

$x = 5 - 3 y$

Step 3.4.2

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

$y = - 3$

$x = 5 - 3 y$

Step 3.4.3

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

$y = 3, - 3$

$x = 5 - 3 y$

$y = 3, - 3$

$x = 5 - 3 y$

$y = 3, - 3$

$x = 5 - 3 y$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = 5 - 3 y$ with $3$ .

$x = 5 - 3 \cdot 3$

$y = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $5 - 3 \cdot 3$ .

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Step 4.2.1.1

Multiply $- 3$ by $3$ .

$x = 5 - 9$

$y = 3$

Step 4.2.1.2

Subtract $9$ from $5$ .

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

Step 5

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

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Step 5.1

Replace all occurrences of $y$ in $x = 5 - 3 y$ with $- 3$ .

$x = 5 - 3 \cdot - 3$

$y = - 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $5 - 3 \cdot - 3$ .

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Step 5.2.1.1

Multiply $- 3$ by $- 3$ .

$x = 5 + 9$

$y = - 3$

Step 5.2.1.2

Add $5$ and $9$ .

$x = 14$

$y = - 3$

$x = 14$

$y = - 3$

$x = 14$

$y = - 3$

$x = 14$

$y = - 3$

Step 6

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

$(- 4, 3)$

$(14, - 3)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 4, 3), (14, - 3)$

Equation Form:

$x = - 4, y = 3$

$x = 14, y = - 3$

Step 8