Trigonometry Examples

$59 {(x + 9)}^{2} + 9 {(y - 19)}^{2} = 3969$

Step 1

Solve the equation for $y$ .

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Step 1.1

Subtract $59 {(x + 9)}^{2}$ from both sides of the equation.

$9 {(y - 19)}^{2} = 3969 - 59 {(x + 9)}^{2}$

Step 1.2

Simplify $3969 - 59 {(x + 9)}^{2}$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite ${(x + 9)}^{2}$ as $(x + 9) (x + 9)$ .

$9 {(y - 19)}^{2} = 3969 - 59 ((x + 9) (x + 9))$

Step 1.2.1.2

Expand $(x + 9) (x + 9)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$9 {(y - 19)}^{2} = 3969 - 59 (x (x + 9) + 9 (x + 9))$

Step 1.2.1.2.2

Apply the distributive property.

$9 {(y - 19)}^{2} = 3969 - 59 (x \cdot x + x \cdot 9 + 9 (x + 9))$

Step 1.2.1.2.3

Apply the distributive property.

$9 {(y - 19)}^{2} = 3969 - 59 (x \cdot x + x \cdot 9 + 9 x + 9 \cdot 9)$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Multiply $x$ by $x$ .

$9 {(y - 19)}^{2} = 3969 - 59 (x^{2} + x \cdot 9 + 9 x + 9 \cdot 9)$

Step 1.2.1.3.1.2

Move $9$ to the left of $x$ .

$9 {(y - 19)}^{2} = 3969 - 59 (x^{2} + 9 \cdot x + 9 x + 9 \cdot 9)$

Step 1.2.1.3.1.3

Multiply $9$ by $9$ .

$9 {(y - 19)}^{2} = 3969 - 59 (x^{2} + 9 x + 9 x + 81)$

Step 1.2.1.3.2

Add $9 x$ and $9 x$ .

$9 {(y - 19)}^{2} = 3969 - 59 (x^{2} + 18 x + 81)$

Step 1.2.1.4

Apply the distributive property.

$9 {(y - 19)}^{2} = 3969 - 59 x^{2} - 59 (18 x) - 59 \cdot 81$

Step 1.2.1.5

Simplify.

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Step 1.2.1.5.1

Multiply $18$ by $- 59$ .

$9 {(y - 19)}^{2} = 3969 - 59 x^{2} - 1062 x - 59 \cdot 81$

Step 1.2.1.5.2

Multiply $- 59$ by $81$ .

$9 {(y - 19)}^{2} = 3969 - 59 x^{2} - 1062 x - 4779$

Step 1.2.2

Subtract $4779$ from $3969$ .

$9 {(y - 19)}^{2} = - 59 x^{2} - 1062 x - 810$

Step 1.3

Divide each term in $9 {(y - 19)}^{2} = - 59 x^{2} - 1062 x - 810$ by $9$ and simplify.

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Step 1.3.1

Divide each term in $9 {(y - 19)}^{2} = - 59 x^{2} - 1062 x - 810$ by $9$ .

$\frac{9 {(y - 19)}^{2}}{9} = \frac{- 59 x^{2}}{9} + \frac{- 1062 x}{9} + \frac{- 810}{9}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $9$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{9 {(y - 19)}^{2}}{9} = \frac{- 59 x^{2}}{9} + \frac{- 1062 x}{9} + \frac{- 810}{9}$

Step 1.3.2.1.2

Divide ${(y - 19)}^{2}$ by $1$ .

${(y - 19)}^{2} = \frac{- 59 x^{2}}{9} + \frac{- 1062 x}{9} + \frac{- 810}{9}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

${(y - 19)}^{2} = - \frac{59 x^{2}}{9} + \frac{- 1062 x}{9} + \frac{- 810}{9}$

Step 1.3.3.1.2

Cancel the common factor of $- 1062$ and $9$ .

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Step 1.3.3.1.2.1

Factor $9$ out of $- 1062 x$ .

${(y - 19)}^{2} = - \frac{59 x^{2}}{9} + \frac{9 (- 118 x)}{9} + \frac{- 810}{9}$

Step 1.3.3.1.2.2

Cancel the common factors.

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Step 1.3.3.1.2.2.1

Factor $9$ out of $9$ .

${(y - 19)}^{2} = - \frac{59 x^{2}}{9} + \frac{9 (- 118 x)}{9 (1)} + \frac{- 810}{9}$

Step 1.3.3.1.2.2.2

Cancel the common factor.

${(y - 19)}^{2} = - \frac{59 x^{2}}{9} + \frac{9 (- 118 x)}{9 \cdot 1} + \frac{- 810}{9}$

Step 1.3.3.1.2.2.3

Rewrite the expression.

${(y - 19)}^{2} = - \frac{59 x^{2}}{9} + \frac{- 118 x}{1} + \frac{- 810}{9}$

Step 1.3.3.1.2.2.4

Divide $- 118 x$ by $1$ .

${(y - 19)}^{2} = - \frac{59 x^{2}}{9} - 118 x + \frac{- 810}{9}$

Step 1.3.3.1.3

Divide $- 810$ by $9$ .

${(y - 19)}^{2} = - \frac{59 x^{2}}{9} - 118 x - 90$

Step 1.4

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

$y - 19 = \pm \sqrt{- \frac{59 x^{2}}{9} - 118 x - 90}$

Step 1.5

Simplify $\pm \sqrt{- \frac{59 x^{2}}{9} - 118 x - 90}$ .

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Step 1.5.1

To write $- 118 x$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y - 19 = \pm \sqrt{- \frac{59 x^{2}}{9} - 118 x \cdot \frac{9}{9} - 90}$

Step 1.5.2

Simplify terms.

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Step 1.5.2.1

Combine $- 118 x$ and $\frac{9}{9}$ .

$y - 19 = \pm \sqrt{- \frac{59 x^{2}}{9} + \frac{- 118 x \cdot 9}{9} - 90}$

Step 1.5.2.2

Combine the numerators over the common denominator.

$y - 19 = \pm \sqrt{\frac{- 59 x^{2} - 118 x \cdot 9}{9} - 90}$

Step 1.5.3

Simplify the numerator.

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Step 1.5.3.1

Factor $59 x$ out of $- 59 x^{2} - 118 x \cdot 9$ .

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Step 1.5.3.1.1

Factor $59 x$ out of $- 59 x^{2}$ .

$y - 19 = \pm \sqrt{\frac{59 x (- x) - 118 x \cdot 9}{9} - 90}$

Step 1.5.3.1.2

Factor $59 x$ out of $- 118 x \cdot 9$ .

$y - 19 = \pm \sqrt{\frac{59 x (- x) + 59 x (- 2 \cdot 9)}{9} - 90}$

Step 1.5.3.1.3

Factor $59 x$ out of $59 x (- x) + 59 x (- 2 \cdot 9)$ .

$y - 19 = \pm \sqrt{\frac{59 x (- x - 2 \cdot 9)}{9} - 90}$

Step 1.5.3.2

Multiply $- 2$ by $9$ .

$y - 19 = \pm \sqrt{\frac{59 x (- x - 18)}{9} - 90}$

Step 1.5.4

To write $- 90$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y - 19 = \pm \sqrt{\frac{59 x (- x - 18)}{9} - 90 \cdot \frac{9}{9}}$

Step 1.5.5

Simplify terms.

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Step 1.5.5.1

Combine $- 90$ and $\frac{9}{9}$ .

$y - 19 = \pm \sqrt{\frac{59 x (- x - 18)}{9} + \frac{- 90 \cdot 9}{9}}$

Step 1.5.5.2

Combine the numerators over the common denominator.

$y - 19 = \pm \sqrt{\frac{59 x (- x - 18) - 90 \cdot 9}{9}}$

Step 1.5.6

Simplify the numerator.

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Step 1.5.6.1

Apply the distributive property.

$y - 19 = \pm \sqrt{\frac{59 x (- x) + 59 x \cdot - 18 - 90 \cdot 9}{9}}$

Step 1.5.6.2

Rewrite using the commutative property of multiplication.

$y - 19 = \pm \sqrt{\frac{59 \cdot - 1 x \cdot x + 59 x \cdot - 18 - 90 \cdot 9}{9}}$

Step 1.5.6.3

Multiply $- 18$ by $59$ .

$y - 19 = \pm \sqrt{\frac{59 \cdot - 1 x \cdot x - 1062 x - 90 \cdot 9}{9}}$

Step 1.5.6.4

Simplify each term.

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Step 1.5.6.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.5.6.4.1.1

Move $x$ .

$y - 19 = \pm \sqrt{\frac{59 \cdot - 1 (x \cdot x) - 1062 x - 90 \cdot 9}{9}}$

Step 1.5.6.4.1.2

Multiply $x$ by $x$ .

$y - 19 = \pm \sqrt{\frac{59 \cdot - 1 x^{2} - 1062 x - 90 \cdot 9}{9}}$

Step 1.5.6.4.2

Multiply $59$ by $- 1$ .

$y - 19 = \pm \sqrt{\frac{- 59 x^{2} - 1062 x - 90 \cdot 9}{9}}$

Step 1.5.6.5

Multiply $- 90$ by $9$ .

$y - 19 = \pm \sqrt{\frac{- 59 x^{2} - 1062 x - 810}{9}}$

Step 1.5.7

Rewrite $\frac{- 59 x^{2} - 1062 x - 810}{9}$ as ${(\frac{1}{3})}^{2} (- 59 x^{2} - 1062 x - 810)$ .

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Step 1.5.7.1

Factor the perfect power $1^{2}$ out of $- 59 x^{2} - 1062 x - 810$ .

$y - 19 = \pm \sqrt{\frac{1^{2} (- 59 x^{2} - 1062 x - 810)}{9}}$

Step 1.5.7.2

Factor the perfect power $3^{2}$ out of $9$ .

$y - 19 = \pm \sqrt{\frac{1^{2} (- 59 x^{2} - 1062 x - 810)}{3^{2} \cdot 1}}$

Step 1.5.7.3

Rearrange the fraction $\frac{1^{2} (- 59 x^{2} - 1062 x - 810)}{3^{2} \cdot 1}$ .

$y - 19 = \pm \sqrt{{(\frac{1}{3})}^{2} (- 59 x^{2} - 1062 x - 810)}$

Step 1.5.8

Pull terms out from under the radical.

$y - 19 = \pm \frac{1}{3} \sqrt{- 59 x^{2} - 1062 x - 810}$

Step 1.5.9

Combine $\frac{1}{3}$ and $\sqrt{- 59 x^{2} - 1062 x - 810}$ .

$y - 19 = \pm \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3}$

Step 1.6

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

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Step 1.6.1

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

$y - 19 = \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3}$

Step 1.6.2

Add $19$ to both sides of the equation.

$y = \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3} + 19$

Step 1.6.3

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

$y - 19 = - \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3}$

Step 1.6.4

Add $19$ to both sides of the equation.

$y = - \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3} + 19$

Step 1.6.5

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

$y = \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3} + 19$

$y = - \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3} + 19$

$y = \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3} + 19$

$y = - \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3} + 19$

$y = \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3} + 19$

$y = - \frac{\sqrt{- 59 x^{2} - 1062 x - 810}}{3} + 19$

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be $0$ or $1$ for each of its variables. In this case, the degree of the variable in the equation violates the linear equation definition, which means that the equation is not a linear equation.

Not Linear