Precalculus Examples

$x^{2} + y^{2} = 65$ , $x^{2} - 3 y = 37$

Step 1

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

$y^{2} = 65 - x^{2}$

$x^{2} - 3 y = 37$

Step 2

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

$y = \pm \sqrt{65 - x^{2}}$

$x^{2} - 3 y = 37$

Step 3

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.

$y = \sqrt{65 - x^{2}}$

$x^{2} - 3 y = 37$

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

$y = - \sqrt{65 - x^{2}}$

$x^{2} - 3 y = 37$

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

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

$x^{2} - 3 y = 37$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

$x^{2} - 3 y = 37$

Step 4

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

$- 3 y = 37 - x^{2}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

Step 5

Divide each term in $- 3 y = 37 - x^{2}$ by $- 3$ and simplify.

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Divide each term in $- 3 y = 37 - x^{2}$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{37}{- 3} + \frac{- x^{2}}{- 3}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

Simplify the left side.

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Cancel the common factor of $- 3$ .

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Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{37}{- 3} + \frac{- x^{2}}{- 3}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

Divide $y$ by $1$ .

$y = \frac{37}{- 3} + \frac{- x^{2}}{- 3}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

$y = \frac{37}{- 3} + \frac{- x^{2}}{- 3}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

$y = \frac{37}{- 3} + \frac{- x^{2}}{- 3}$

$y = \sqrt{65 - x^{2}}, - \sqrt{65 - x^{2}}$

Simplify the right side.

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Simplify each term.

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Move the negative in front of the fraction.

$y = - \frac{37}{3} + \frac{- x^{2}}{- 3}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

Dividing two negative values results in a positive value.

$y = - \frac{37}{3} + \frac{x^{2}}{3}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

$y = - \frac{37}{3} + \frac{x^{2}}{3}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

$y = - \frac{37}{3} + \frac{x^{2}}{3}$

$y = \sqrt{65 - x^{2}}, - \sqrt{65 - x^{2}}$

$y = - \frac{37}{3} + \frac{x^{2}}{3}$

$y = \sqrt{65 - x^{2}}$

$y = - \sqrt{65 - x^{2}}$

Step 6

Create a graph to locate the intersection of the equations. The intersection of the system of equations is the solution.

$(7, 4)$

$(- 7, 4)$

$(4, - 7)$

$(- 4, - 7)$

Step 7