Algebra Examples

$2 x - 3 y = 5$

Step 1

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

$2 (r \cos (θ)) - 3 y = 5$

Step 2

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

$2 (r \cos (θ)) - 3 (r \sin (θ)) = 5$

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.

$2 r \cos (θ) - 3 r \sin (θ) = 5$

Step 3.2

Factor $r$ out of $2 r \cos (θ) - 3 r \sin (θ)$ .

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Step 3.2.1

Factor $r$ out of $2 r \cos (θ)$ .

$r (2 \cos (θ)) - 3 r \sin (θ) = 5$

Step 3.2.2

Factor $r$ out of $- 3 r \sin (θ)$ .

$r (2 \cos (θ)) + r (- 3 \sin (θ)) = 5$

Step 3.2.3

Factor $r$ out of $r (2 \cos (θ)) + r (- 3 \sin (θ))$ .

$r (2 \cos (θ) - 3 \sin (θ)) = 5$

Step 3.3

Divide each term in $r (2 \cos (θ) - 3 \sin (θ)) = 5$ by $2 \cos (θ) - 3 \sin (θ)$ and simplify.

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Step 3.3.1

Divide each term in $r (2 \cos (θ) - 3 \sin (θ)) = 5$ by $2 \cos (θ) - 3 \sin (θ)$ .

$\frac{r (2 \cos (θ) - 3 \sin (θ))}{2 \cos (θ) - 3 \sin (θ)} = \frac{5}{2 \cos (θ) - 3 \sin (θ)}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2 \cos (θ) - 3 \sin (θ)$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{r (2 \cos (θ) - 3 \sin (θ))}{2 \cos (θ) - 3 \sin (θ)} = \frac{5}{2 \cos (θ) - 3 \sin (θ)}$

Step 3.3.2.1.2

Divide $r$ by $1$ .

$r = \frac{5}{2 \cos (θ) - 3 \sin (θ)}$