Precalculus Examples

$x^{2} + y^{2} = 49$ , $y = x - 3$

Step 1

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

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Step 1.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 49$ with $x - 3$ .

$x^{2} + {(x - 3)}^{2} = 49$

$y = x - 3$

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

$x^{2} + (x - 3) (x - 3) = 49$

$y = x - 3$

Step 1.2.1.1.2

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

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Step 1.2.1.1.2.1

Apply the distributive property.

$x^{2} + x (x - 3) - 3 (x - 3) = 49$

$y = x - 3$

Step 1.2.1.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot - 3 - 3 (x - 3) = 49$

$y = x - 3$

Step 1.2.1.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 = 49$

$y = x - 3$

$x^{2} + x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 = 49$

$y = x - 3$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 = 49$

$y = x - 3$

Step 1.2.1.1.3.1.2

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

$x^{2} + x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 = 49$

$y = x - 3$

Step 1.2.1.1.3.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} + x^{2} - 3 x - 3 x + 9 = 49$

$y = x - 3$

$x^{2} + x^{2} - 3 x - 3 x + 9 = 49$

$y = x - 3$

Step 1.2.1.1.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} + x^{2} - 6 x + 9 = 49$

$y = x - 3$

$x^{2} + x^{2} - 6 x + 9 = 49$

$y = x - 3$

$x^{2} + x^{2} - 6 x + 9 = 49$

$y = x - 3$

Step 1.2.1.2

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

$2 x^{2} - 6 x + 9 = 49$

$y = x - 3$

$2 x^{2} - 6 x + 9 = 49$

$y = x - 3$

$2 x^{2} - 6 x + 9 = 49$

$y = x - 3$

$2 x^{2} - 6 x + 9 = 49$

$y = x - 3$

Step 2

Solve for $x$ in $2 x^{2} - 6 x + 9 = 49$ .

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Step 2.1

Subtract $49$ from both sides of the equation.

$2 x^{2} - 6 x + 9 - 49 = 0$

$y = x - 3$

Step 2.2

Subtract $49$ from $9$ .

$2 x^{2} - 6 x - 40 = 0$

$y = x - 3$

Step 2.3

Factor $2$ out of $2 x^{2} - 6 x - 40$ .

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Step 2.3.1

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

$2 (x^{2}) - 6 x - 40 = 0$

$y = x - 3$

Step 2.3.2

Factor $2$ out of $- 6 x$ .

$2 (x^{2}) + 2 (- 3 x) - 40 = 0$

$y = x - 3$

Step 2.3.3

Factor $2$ out of $- 40$ .

$2 x^{2} + 2 (- 3 x) + 2 \cdot - 20 = 0$

$y = x - 3$

Step 2.3.4

Factor $2$ out of $2 x^{2} + 2 (- 3 x)$ .

$2 (x^{2} - 3 x) + 2 \cdot - 20 = 0$

$y = x - 3$

Step 2.3.5

Factor $2$ out of $2 (x^{2} - 3 x) + 2 \cdot - 20$ .

$2 (x^{2} - 3 x - 20) = 0$

$y = x - 3$

$2 (x^{2} - 3 x - 20) = 0$

$y = x - 3$

Step 2.4

Divide each term in $2 (x^{2} - 3 x - 20) = 0$ by $2$ and simplify.

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Step 2.4.1

Divide each term in $2 (x^{2} - 3 x - 20) = 0$ by $2$ .

$\frac{2 (x^{2} - 3 x - 20)}{2} = \frac{0}{2}$

$y = x - 3$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - 3 x - 20)}{2} = \frac{0}{2}$

$y = x - 3$

Step 2.4.2.1.2

Divide $x^{2} - 3 x - 20$ by $1$ .

$x^{2} - 3 x - 20 = \frac{0}{2}$

$y = x - 3$

$x^{2} - 3 x - 20 = \frac{0}{2}$

$y = x - 3$

$x^{2} - 3 x - 20 = \frac{0}{2}$

$y = x - 3$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $0$ by $2$ .

$x^{2} - 3 x - 20 = 0$

$y = x - 3$

$x^{2} - 3 x - 20 = 0$

$y = x - 3$

$x^{2} - 3 x - 20 = 0$

$y = x - 3$

Step 2.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$y = x - 3$

Step 2.6

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 20)}}{2 \cdot 1}$

$y = x - 3$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

$y = x - 3$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 20}}{2 \cdot 1}$

$y = x - 3$

Step 2.7.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{3 \pm \sqrt{9 + 80}}{2 \cdot 1}$

$y = x - 3$

$x = \frac{3 \pm \sqrt{9 + 80}}{2 \cdot 1}$

$y = x - 3$

Step 2.7.1.3

Add $9$ and $80$ .

$x = \frac{3 \pm \sqrt{89}}{2 \cdot 1}$

$y = x - 3$

$x = \frac{3 \pm \sqrt{89}}{2 \cdot 1}$

$y = x - 3$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{89}}{2}$

$y = x - 3$

$x = \frac{3 \pm \sqrt{89}}{2}$

$y = x - 3$

Step 2.8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

$y = x - 3$

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 20}}{2 \cdot 1}$

$y = x - 3$

Step 2.8.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{3 \pm \sqrt{9 + 80}}{2 \cdot 1}$

$y = x - 3$

$x = \frac{3 \pm \sqrt{9 + 80}}{2 \cdot 1}$

$y = x - 3$

Step 2.8.1.3

Add $9$ and $80$ .

$x = \frac{3 \pm \sqrt{89}}{2 \cdot 1}$

$y = x - 3$

$x = \frac{3 \pm \sqrt{89}}{2 \cdot 1}$

$y = x - 3$

Step 2.8.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{89}}{2}$

$y = x - 3$

Step 2.8.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{89}}{2}$

$y = x - 3$

$x = \frac{3 + \sqrt{89}}{2}$

$y = x - 3$

Step 2.9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.9.1

Simplify the numerator.

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Step 2.9.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

$y = x - 3$

Step 2.9.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 2.9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 20}}{2 \cdot 1}$

$y = x - 3$

Step 2.9.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{3 \pm \sqrt{9 + 80}}{2 \cdot 1}$

$y = x - 3$

$x = \frac{3 \pm \sqrt{9 + 80}}{2 \cdot 1}$

$y = x - 3$

Step 2.9.1.3

Add $9$ and $80$ .

$x = \frac{3 \pm \sqrt{89}}{2 \cdot 1}$

$y = x - 3$

$x = \frac{3 \pm \sqrt{89}}{2 \cdot 1}$

$y = x - 3$

Step 2.9.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{89}}{2}$

$y = x - 3$

Step 2.9.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{89}}{2}$

$y = x - 3$

$x = \frac{3 - \sqrt{89}}{2}$

$y = x - 3$

Step 2.10

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{89}}{2}, \frac{3 - \sqrt{89}}{2}$

$y = x - 3$

$x = \frac{3 + \sqrt{89}}{2}, \frac{3 - \sqrt{89}}{2}$

$y = x - 3$

Step 3

Replace all occurrences of $x$ with $\frac{3 + \sqrt{89}}{2}$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = x - 3$ with $\frac{3 + \sqrt{89}}{2}$ .

$y = (\frac{3 + \sqrt{89}}{2}) - 3$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $(\frac{3 + \sqrt{89}}{2}) - 3$ .

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Step 3.2.1.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{3 + \sqrt{89}}{2} - 3 \cdot \frac{2}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2.1.2

Combine fractions.

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Step 3.2.1.2.1

Combine $- 3$ and $\frac{2}{2}$ .

$y = \frac{3 + \sqrt{89}}{2} + \frac{- 3 \cdot 2}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2.1.2.2

Combine the numerators over the common denominator.

$y = \frac{3 + \sqrt{89} - 3 \cdot 2}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

$y = \frac{3 + \sqrt{89} - 3 \cdot 2}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2.1.3

Simplify the numerator.

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Step 3.2.1.3.1

Multiply $- 3$ by $2$ .

$y = \frac{3 + \sqrt{89} - 6}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2.1.3.2

Subtract $6$ from $3$ .

$y = \frac{- 3 + \sqrt{89}}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

$y = \frac{- 3 + \sqrt{89}}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2.1.4

Simplify with factoring out.

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Step 3.2.1.4.1

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{- 1 \cdot 3 + \sqrt{89}}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2.1.4.2

Factor $- 1$ out of $\sqrt{89}$ .

$y = \frac{- 1 \cdot 3 - 1 (- \sqrt{89})}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2.1.4.3

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{89})$ .

$y = \frac{- 1 (3 - \sqrt{89})}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 3.2.1.4.4

Move the negative in front of the fraction.

$y = - \frac{3 - \sqrt{89}}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

$y = - \frac{3 - \sqrt{89}}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

$y = - \frac{3 - \sqrt{89}}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

$y = - \frac{3 - \sqrt{89}}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

$y = - \frac{3 - \sqrt{89}}{2}$

$x = \frac{3 + \sqrt{89}}{2}$

Step 4

Replace all occurrences of $x$ with $\frac{3 - \sqrt{89}}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = x - 3$ with $\frac{3 - \sqrt{89}}{2}$ .

$y = (\frac{3 - \sqrt{89}}{2}) - 3$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $(\frac{3 - \sqrt{89}}{2}) - 3$ .

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Step 4.2.1.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{3 - \sqrt{89}}{2} - 3 \cdot \frac{2}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine $- 3$ and $\frac{2}{2}$ .

$y = \frac{3 - \sqrt{89}}{2} + \frac{- 3 \cdot 2}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2.1.2.2

Combine the numerators over the common denominator.

$y = \frac{3 - \sqrt{89} - 3 \cdot 2}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

$y = \frac{3 - \sqrt{89} - 3 \cdot 2}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2.1.3

Simplify the numerator.

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Step 4.2.1.3.1

Multiply $- 3$ by $2$ .

$y = \frac{3 - \sqrt{89} - 6}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2.1.3.2

Subtract $6$ from $3$ .

$y = \frac{- 3 - \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

$y = \frac{- 3 - \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2.1.4

Simplify with factoring out.

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Step 4.2.1.4.1

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{- 1 \cdot 3 - \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2.1.4.2

Factor $- 1$ out of $- \sqrt{89}$ .

$y = \frac{- 1 \cdot 3 - (\sqrt{89})}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2.1.4.3

Factor $- 1$ out of $- 1 (3) - (\sqrt{89})$ .

$y = \frac{- 1 (3 + \sqrt{89})}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 4.2.1.4.4

Move the negative in front of the fraction.

$y = - \frac{3 + \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

$y = - \frac{3 + \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

$y = - \frac{3 + \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

$y = - \frac{3 + \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

$y = - \frac{3 + \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}$

Step 5

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

$(\frac{3 + \sqrt{89}}{2}, - \frac{3 - \sqrt{89}}{2})$

$(\frac{3 - \sqrt{89}}{2}, - \frac{3 + \sqrt{89}}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{3 + \sqrt{89}}{2}, - \frac{3 - \sqrt{89}}{2}), (\frac{3 - \sqrt{89}}{2}, - \frac{3 + \sqrt{89}}{2})$

Equation Form:

$x = \frac{3 + \sqrt{89}}{2}, y = - \frac{3 - \sqrt{89}}{2}$

$x = \frac{3 - \sqrt{89}}{2}, y = - \frac{3 + \sqrt{89}}{2}$

Step 7