Precalculus Examples

$- x + y = 3$ , $x^{2} - 6 x - 27 + y^{2} = 0$

Step 1

Add $x$ to both sides of the equation.

$y = 3 + x$

$x^{2} - 6 x - 27 + y^{2} = 0$

Step 2

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.

$- 6 x - 27 + y^{2} = - x^{2}$

$y = 3 + x$

Add $6 x$ to both sides of the equation.

$- 27 + y^{2} = - x^{2} + 6 x$

$y = 3 + x$

Add $27$ to both sides of the equation.

$y^{2} = - x^{2} + 6 x + 27$

$y = 3 + x$

$y^{2} = - x^{2} + 6 x + 27$

$y = 3 + x$

Step 3

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

$y = \pm \sqrt{- x^{2} + 6 x + 27}$

$y = 3 + x$

Step 4

Factor by grouping.

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For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 27 = - 27$ and whose sum is $b = 6$ .

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Factor $6$ out of $6 x$ .

$y = \pm \sqrt{- x^{2} + 6 (x) + 27}$

$y = 3 + x$

Rewrite $6$ as $- 3$ plus $9$

$y = \pm \sqrt{- x^{2} + (- 3 + 9) x + 27}$

$y = 3 + x$

Apply the distributive property.

$y = \pm \sqrt{- x^{2} - 3 x + 9 x + 27}$

$y = 3 + x$

$y = \pm \sqrt{- x^{2} - 3 x + 9 x + 27}$

$y = 3 + x$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$y = \pm \sqrt{(- x^{2} - 3 x) + 9 x + 27}$

$y = 3 + x$

Factor out the greatest common factor (GCF) from each group.

$y = \pm \sqrt{x (- x - 3) - 9 (- x - 3)}$

$y = 3 + x$

$y = \pm \sqrt{x (- x - 3) - 9 (- x - 3)}$

$y = 3 + x$

Factor the polynomial by factoring out the greatest common factor, $- x - 3$ .

$y = \pm \sqrt{(- x - 3) (x - 9)}$

$y = 3 + x$

$y = \pm \sqrt{(- x - 3) (x - 9)}$

$y = 3 + x$

Step 5

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.

$y = \sqrt{(- x - 3) (x - 9)}$

$y = 3 + x$

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

$y = - \sqrt{(- x - 3) (x - 9)}$

$y = 3 + x$

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

$y = \sqrt{(- x - 3) (x - 9)}$

$y = - \sqrt{(- x - 3) (x - 9)}$

$y = 3 + x$

$y = \sqrt{(- x - 3) (x - 9)}$

$y = - \sqrt{(- x - 3) (x - 9)}$

$y = 3 + x$

Step 6

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

$(3, 6)$

$(- 3, 0)$

Step 7