Algebra Examples

$r = \frac{6}{2 - 3 \sin (θ)}$

Step 1

Since $\sin (θ) = \frac{y}{r}$ , replace $\sin (θ)$ with $\frac{y}{r}$ .

$r = \frac{6}{2 - 3 \frac{y}{r}}$

Step 2

Multiply both sides by $r$ .

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Step 2.1

Multiply both sides by $r$ .

$r \cdot r = r \frac{6}{2 - 3 \frac{y}{r}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Multiply $r$ by $r$ .

$r^{2} = r \frac{6}{2 - 3 \frac{y}{r}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify $r \frac{6}{2 - 3 \frac{y}{r}}$ .

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Step 2.3.1.1

Simplify the denominator.

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Step 2.3.1.1.1

Combine $- 3$ and $\frac{y}{r}$ .

$r^{2} = r \frac{6}{2 + \frac{- 3 y}{r}}$

Step 2.3.1.1.2

Move the negative in front of the fraction.

$r^{2} = r \frac{6}{2 - \frac{3 y}{r}}$

Step 2.3.1.1.3

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

$r^{2} = r \frac{6}{\frac{2 r}{r} - \frac{3 y}{r}}$

Step 2.3.1.1.4

Combine the numerators over the common denominator.

$r^{2} = r \frac{6}{\frac{2 r - 3 y}{r}}$

Step 2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$r^{2} = r (6 \frac{r}{2 r - 3 y})$

Step 2.3.1.3

Rewrite using the commutative property of multiplication.

$r^{2} = 6 r \frac{r}{2 r - 3 y}$

Step 2.3.1.4

Multiply $6 r \frac{r}{2 r - 3 y}$ .

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Step 2.3.1.4.1

Combine $6$ and $\frac{r}{2 r - 3 y}$ .

$r^{2} = r \frac{6 r}{2 r - 3 y}$

Step 2.3.1.4.2

Combine $r$ and $\frac{6 r}{2 r - 3 y}$ .

$r^{2} = \frac{r (6 r)}{2 r - 3 y}$

Step 2.3.1.4.3

Raise $r$ to the power of $1$ .

$r^{2} = \frac{6 (r^{1} r)}{2 r - 3 y}$

Step 2.3.1.4.4

Raise $r$ to the power of $1$ .

$r^{2} = \frac{6 (r^{1} r^{1})}{2 r - 3 y}$

Step 2.3.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r^{2} = \frac{6 r^{1 + 1}}{2 r - 3 y}$

Step 2.3.1.4.6

Add $1$ and $1$ .

$r^{2} = \frac{6 r^{2}}{2 r - 3 y}$

Step 3

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = \frac{6 (x^{2} + y^{2})}{2 \sqrt{x^{2} + y^{2}} - 3 y}$