Algebra Examples

$3 x - y + 2 = 0$

Step 1

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

$3 (r \cos (θ)) - y + 2 = 0$

Step 2

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

$3 (r \cos (θ)) - (r \sin (θ)) + 2 = 0$

Step 3

Solve for $r$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Remove parentheses.

$3 r \cos (θ) - r \sin (θ) + 2 = 0$

Step 3.2

Subtract $2$ from both sides of the equation.

$3 r \cos (θ) - r \sin (θ) = - 2$

Step 3.3

Factor $r$ out of $3 r \cos (θ) - r \sin (θ)$ .

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Step 3.3.1

Factor $r$ out of $3 r \cos (θ)$ .

$r (3 \cos (θ)) - r \sin (θ) = - 2$

Step 3.3.2

Factor $r$ out of $- r \sin (θ)$ .

$r (3 \cos (θ)) + r (- 1 \sin (θ)) = - 2$

Step 3.3.3

Factor $r$ out of $r (3 \cos (θ)) + r (- 1 \sin (θ))$ .

$r (3 \cos (θ) - 1 \sin (θ)) = - 2$

Step 3.4

Rewrite $- 1 \sin (θ)$ as $- \sin (θ)$ .

$r (3 \cos (θ) - \sin (θ)) = - 2$

Step 3.5

Divide each term in $r (3 \cos (θ) - \sin (θ)) = - 2$ by $3 \cos (θ) - \sin (θ)$ and simplify.

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Step 3.5.1

Divide each term in $r (3 \cos (θ) - \sin (θ)) = - 2$ by $3 \cos (θ) - \sin (θ)$ .

$\frac{r (3 \cos (θ) - \sin (θ))}{3 \cos (θ) - \sin (θ)} = \frac{- 2}{3 \cos (θ) - \sin (θ)}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $3 \cos (θ) - \sin (θ)$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{r (3 \cos (θ) - \sin (θ))}{3 \cos (θ) - \sin (θ)} = \frac{- 2}{3 \cos (θ) - \sin (θ)}$

Step 3.5.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 2}{3 \cos (θ) - \sin (θ)}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Move the negative in front of the fraction.

$r = - \frac{2}{3 \cos (θ) - \sin (θ)}$