Algebra Examples

$x = \frac{y^{2}}{2}$

Step 1

Since $x = r \cos (θ)$ , replace $x$ with $r \cos (θ)$ .

$r \cos (θ) = \frac{y^{2}}{2}$

Step 2

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

$r \cos (θ) = \frac{{(r \sin (θ))}^{2}}{2}$

Step 3

Solve for $r$ .

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Step 3.1

Multiply both sides by $2$ .

$r \cos (θ) \cdot 2 = \frac{{(r \sin (θ))}^{2}}{2} \cdot 2$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Move $2$ to the left of $r \cos (θ)$ .

$2 r \cos (θ) = \frac{{(r \sin (θ))}^{2}}{2} \cdot 2$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{{(r \sin (θ))}^{2}}{2} \cdot 2$ .

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Step 3.2.2.1.1

Apply the product rule to $r \sin (θ)$ .

$2 r \cos (θ) = \frac{r^{2} \sin^{2} (θ)}{2} \cdot 2$

Step 3.2.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.2.1

Cancel the common factor.

$2 r \cos (θ) = \frac{r^{2} \sin^{2} (θ)}{2} \cdot 2$

Step 3.2.2.1.2.2

Rewrite the expression.

$2 r \cos (θ) = r^{2} \sin^{2} (θ)$

Step 3.3

Solve for $r$ .

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Step 3.3.1

Subtract $r^{2} \sin^{2} (θ)$ from both sides of the equation.

$2 r \cos (θ) - r^{2} \sin^{2} (θ) = 0$

Step 3.3.2

Factor $r$ out of $2 r \cos (θ) - r^{2} \sin^{2} (θ)$ .

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Step 3.3.2.1

Factor $r$ out of $2 r \cos (θ)$ .

$r (2 \cos (θ)) - r^{2} \sin^{2} (θ) = 0$

Step 3.3.2.2

Factor $r$ out of $- r^{2} \sin^{2} (θ)$ .

$r (2 \cos (θ)) + r (- r \sin^{2} (θ)) = 0$

Step 3.3.2.3

Factor $r$ out of $r (2 \cos (θ)) + r (- r \sin^{2} (θ))$ .

$r (2 \cos (θ) - r \sin^{2} (θ)) = 0$

Step 3.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$r = 0$

$2 \cos (θ) - r \sin^{2} (θ) = 0$

Step 3.3.4

Set $r$ equal to $0$ .

$r = 0$

Step 3.3.5

Set $2 \cos (θ) - r \sin^{2} (θ)$ equal to $0$ and solve for $r$ .

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Step 3.3.5.1

Set $2 \cos (θ) - r \sin^{2} (θ)$ equal to $0$ .

$2 \cos (θ) - r \sin^{2} (θ) = 0$

Step 3.3.5.2

Solve $2 \cos (θ) - r \sin^{2} (θ) = 0$ for $r$ .

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Step 3.3.5.2.1

Subtract $2 \cos (θ)$ from both sides of the equation.

$- r \sin^{2} (θ) = - 2 \cos (θ)$

Step 3.3.5.2.2

Divide each term in $- r \sin^{2} (θ) = - 2 \cos (θ)$ by $- \sin^{2} (θ)$ and simplify.

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Step 3.3.5.2.2.1

Divide each term in $- r \sin^{2} (θ) = - 2 \cos (θ)$ by $- \sin^{2} (θ)$ .

$\frac{- r \sin^{2} (θ)}{- \sin^{2} (θ)} = \frac{- 2 \cos (θ)}{- \sin^{2} (θ)}$

Step 3.3.5.2.2.2

Simplify the left side.

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Step 3.3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{r \sin^{2} (θ)}{\sin^{2} (θ)} = \frac{- 2 \cos (θ)}{- \sin^{2} (θ)}$

Step 3.3.5.2.2.2.2

Cancel the common factor of $\sin^{2} (θ)$ .

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Step 3.3.5.2.2.2.2.1

Cancel the common factor.

$\frac{r \sin^{2} (θ)}{\sin^{2} (θ)} = \frac{- 2 \cos (θ)}{- \sin^{2} (θ)}$

Step 3.3.5.2.2.2.2.2

Divide $r$ by $1$ .

$r = \frac{- 2 \cos (θ)}{- \sin^{2} (θ)}$

Step 3.3.5.2.2.3

Simplify the right side.

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Step 3.3.5.2.2.3.1

Factor $\sin (θ)$ out of $\sin^{2} (θ)$ .

$r = \frac{- 2 \cos (θ)}{- (\sin (θ) \sin (θ))}$

Step 3.3.5.2.2.3.2

Separate fractions.

$r = \frac{- 2}{- (\sin (θ))} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 3.3.5.2.2.3.3

Convert from $\frac{\cos (θ)}{\sin (θ)}$ to $\cot (θ)$ .

$r = \frac{- 2}{- (\sin (θ))} \cot (θ)$

Step 3.3.5.2.2.3.4

Separate fractions.

$r = \frac{- 2}{- 1} \cdot \frac{1}{\sin (θ)} \cot (θ)$

Step 3.3.5.2.2.3.5

Convert from $\frac{1}{\sin (θ)}$ to $\csc (θ)$ .

$r = \frac{- 2}{- 1} \csc (θ) \cot (θ)$

Step 3.3.5.2.2.3.6

Divide $- 2$ by $- 1$ .

$r = 2 \csc (θ) \cot (θ)$

Step 3.3.6

The final solution is all the values that make $r (2 \cos (θ) - r \sin^{2} (θ)) = 0$ true.

$r = 0$

$r = 2 \csc (θ) \cot (θ)$

$r = 0$

$r = 2 \csc (θ) \cot (θ)$

$r = 0$

$r = 2 \csc (θ) \cot (θ)$