Precalculus Examples

$2 p x + 2 p y = 26$ , $p x^{2} + p y^{2} = 89 p$

Step 1

Solve the equation for $p$ .

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Factor $2 p$ out of $2 p x + 2 p y$ .

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Factor $2 p$ out of $2 p x$ .

$2 p (x) + 2 p y = 26$

$p x^{2} + p y^{2} = 89 p$

Factor $2 p$ out of $2 p y$ .

$2 p (x) + 2 p (y) = 26$

$p x^{2} + p y^{2} = 89 p$

Factor $2 p$ out of $2 p (x) + 2 p (y)$ .

$2 p (x + y) = 26$

$p x^{2} + p y^{2} = 89 p$

$2 p (x + y) = 26$

$p x^{2} + p y^{2} = 89 p$

Divide each term in $2 p (x + y) = 26$ by $2 (x + y)$ and simplify.

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Divide each term in $2 p (x + y) = 26$ by $2 (x + y)$ .

$\frac{2 p (x + y)}{2 (x + y)} = \frac{26}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 p (x + y)}{2 (x + y)} = \frac{26}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

Rewrite the expression.

$\frac{p (x + y)}{x + y} = \frac{26}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

$\frac{p (x + y)}{x + y} = \frac{26}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

Cancel the common factor of $x + y$ .

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Cancel the common factor.

$\frac{p (x + y)}{x + y} = \frac{26}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

Divide $p$ by $1$ .

$p = \frac{26}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

$p = \frac{26}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

$p = \frac{26}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

Simplify the right side.

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Cancel the common factor of $26$ and $2$ .

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Factor $2$ out of $26$ .

$p = \frac{2 \cdot 13}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

Cancel the common factors.

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Cancel the common factor.

$p = \frac{2 \cdot 13}{2 (x + y)}$

$p x^{2} + p y^{2} = 89 p$

Rewrite the expression.

$p = \frac{13}{x + y}$

$p x^{2} + p y^{2} = 89 p$

$p = \frac{13}{x + y}$

$p x^{2} + p y^{2} = 89 p$

$p = \frac{13}{x + y}$

$p x^{2} + p y^{2} = 89 p$

$p = \frac{13}{x + y}$

$p x^{2} + p y^{2} = 89 p$

$p = \frac{13}{x + y}$

$p x^{2} + p y^{2} = 89 p$

$p = \frac{13}{x + y}$

$p x^{2} + p y^{2} = 89 p$

Step 2

Solve the equation for $y$ .

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Simplify $(\frac{13}{x + y}) \cdot x^{2} + (\frac{13}{x + y}) \cdot y^{2}$ .

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Simplify each term.

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Combine $\frac{13}{x + y}$ and $x^{2}$ .

$\frac{13 x^{2}}{x + y} + (\frac{13}{x + y}) \cdot y^{2} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

Combine $\frac{13}{x + y}$ and $y^{2}$ .

$\frac{13 x^{2}}{x + y} + \frac{13 y^{2}}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

$\frac{13 x^{2}}{x + y} + \frac{13 y^{2}}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

Simplify terms.

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Combine the numerators over the common denominator.

$\frac{13 x^{2} + 13 y^{2}}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

Factor $13$ out of $13 x^{2} + 13 y^{2}$ .

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Factor $13$ out of $13 x^{2}$ .

$\frac{13 (x^{2}) + 13 y^{2}}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

Factor $13$ out of $13 y^{2}$ .

$\frac{13 (x^{2}) + 13 (y^{2})}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

Factor $13$ out of $13 (x^{2}) + 13 (y^{2})$ .

$\frac{13 (x^{2} + y^{2})}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

$\frac{13 (x^{2} + y^{2})}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

$\frac{13 (x^{2} + y^{2})}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

$\frac{13 (x^{2} + y^{2})}{x + y} = 89 (\frac{13}{x + y})$

$p = \frac{13}{x + y}$

Multiply $89 (\frac{13}{x + y})$ .

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Combine $89$ and $\frac{13}{x + y}$ .

$\frac{13 (x^{2} + y^{2})}{x + y} = \frac{89 \cdot 13}{x + y}$

$p = \frac{13}{x + y}$

Multiply $89$ by $13$ .

$\frac{13 (x^{2} + y^{2})}{x + y} = \frac{1157}{x + y}$

$p = \frac{13}{x + y}$

$\frac{13 (x^{2} + y^{2})}{x + y} = \frac{1157}{x + y}$

$p = \frac{13}{x + y}$

Move all terms containing $y$ to the left side of the equation.

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Subtract $\frac{1157}{x + y}$ from both sides of the equation.

$\frac{13 (x^{2} + y^{2})}{x + y} - \frac{1157}{x + y} = 0$

$p = \frac{13}{x + y}$

Combine the numerators over the common denominator.

$\frac{13 (x^{2} + y^{2}) - 1157}{x + y} = 0$

$p = \frac{13}{x + y}$

Factor $13$ out of $13 (x^{2} + y^{2}) - 1157$ .

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Factor $13$ out of $- 1157$ .

$\frac{13 (x^{2} + y^{2}) + 13 \cdot - 89}{x + y} = 0$

$p = \frac{13}{x + y}$

Factor $13$ out of $13 (x^{2} + y^{2}) + 13 \cdot - 89$ .

$\frac{13 (x^{2} + y^{2} - 89)}{x + y} = 0$

$p = \frac{13}{x + y}$

$\frac{13 (x^{2} + y^{2} - 89)}{x + y} = 0$

$p = \frac{13}{x + y}$

$\frac{13 (x^{2} + y^{2} - 89)}{x + y} = 0$

$p = \frac{13}{x + y}$

Set the numerator equal to zero.

$13 (x^{2} + y^{2} - 89) = 0$

$p = \frac{13}{x + y}$

Solve the equation for $y$ .

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Divide each term in $13 (x^{2} + y^{2} - 89) = 0$ by $13$ and simplify.

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Divide each term in $13 (x^{2} + y^{2} - 89) = 0$ by $13$ .

$\frac{13 (x^{2} + y^{2} - 89)}{13} = \frac{0}{13}$

$p = \frac{13}{x + y}$

Simplify the left side.

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Cancel the common factor of $13$ .

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Cancel the common factor.

$\frac{13 (x^{2} + y^{2} - 89)}{13} = \frac{0}{13}$

$p = \frac{13}{x + y}$

Divide $x^{2} + y^{2} - 89$ by $1$ .

$x^{2} + y^{2} - 89 = \frac{0}{13}$

$p = \frac{13}{x + y}$

$x^{2} + y^{2} - 89 = \frac{0}{13}$

$p = \frac{13}{x + y}$

$x^{2} + y^{2} - 89 = \frac{0}{13}$

$p = \frac{13}{x + y}$

Simplify the right side.

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Divide $0$ by $13$ .

$x^{2} + y^{2} - 89 = 0$

$p = \frac{13}{x + y}$

$x^{2} + y^{2} - 89 = 0$

$p = \frac{13}{x + y}$

$x^{2} + y^{2} - 89 = 0$

$p = \frac{13}{x + y}$

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

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

$y^{2} - 89 = - x^{2}$

$p = \frac{13}{x + y}$

Add $89$ to both sides of the equation.

$y^{2} = - x^{2} + 89$

$p = \frac{13}{x + y}$

$y^{2} = - x^{2} + 89$

$p = \frac{13}{x + y}$

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

$y = \pm \sqrt{- x^{2} + 89}$

$p = \frac{13}{x + y}$

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

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

$y = \sqrt{- x^{2} + 89}$

$p = \frac{13}{x + y}$

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

$y = - \sqrt{- x^{2} + 89}$

$p = \frac{13}{x + y}$

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

$y = \sqrt{- x^{2} + 89}$

$y = - \sqrt{- x^{2} + 89}$

$p = \frac{13}{x + y}$

$y = \sqrt{- x^{2} + 89}$

$y = - \sqrt{- x^{2} + 89}$

$p = \frac{13}{x + y}$

$y = \sqrt{- x^{2} + 89}$

$y = - \sqrt{- x^{2} + 89}$

$p = \frac{13}{x + y}$

$y = \sqrt{- x^{2} + 89}$

$y = - \sqrt{- x^{2} + 89}$

$p = \frac{13}{x + y}$

Step 3

The simplified system is the arbitrary solution of the original system of equations.

$p = \frac{13}{x + (\sqrt{- x^{2} + 89}, - \sqrt{- x^{2} + 89})}$

$y = \sqrt{- x^{2} + 89}$

$y = - \sqrt{- x^{2} + 89}$