Algebra Examples

$r = \sin (θ) - \cos (θ)$

Step 1

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

$r = \frac{y}{r} - \cos (θ)$

Step 2

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

$r = \frac{y}{r} - \frac{x}{r}$

Step 3

Multiply both sides by $r$ .

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Step 3.1

Multiply both sides by $r$ .

$r \cdot r = r (\frac{y}{r} - \frac{x}{r})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $r$ by $r$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $r (\frac{y}{r} - \frac{x}{r})$ .

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Step 3.3.1.1

Apply the distributive property.

$r^{2} = r \frac{y}{r} + r (- \frac{x}{r})$

Step 3.3.1.2

Cancel the common factor of $r$ .

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Step 3.3.1.2.1

Cancel the common factor.

$r^{2} = r \frac{y}{r} + r (- \frac{x}{r})$

Step 3.3.1.2.2

Rewrite the expression.

$r^{2} = y + r (- \frac{x}{r})$

Step 3.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.4

Cancel the common factor of $r$ .

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Step 3.3.1.4.1

Factor $r$ out of $- r$ .

$r^{2} = y + r \cdot - 1 \frac{x}{r}$

Step 3.3.1.4.2

Cancel the common factor.

$r^{2} = y + r \cdot - 1 \frac{x}{r}$

Step 3.3.1.4.3

Rewrite the expression.

$r^{2} = y - x$

Step 4

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

$x^{2} + y^{2} = y - x$

Step 5

Write in standard form.

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Step 5.1

Move all terms containing variables to the left side of the equation.

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Step 5.1.1

Subtract $y$ from both sides of the equation.

$x^{2} + y^{2} - y = - x$

Step 5.1.2

Add $x$ to both sides of the equation.

$x^{2} + y^{2} - y + x = 0$

Step 5.1.3

Move $- y$ .

$x^{2} + y^{2} + x - y = 0$

Step 5.2

Complete the square for $x^{2} + x$ .

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Step 5.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 1$

$c = 0$

Step 5.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{1}{2 \cdot 1}$

Step 5.2.3.2

Cancel the common factor of $1$ .

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Step 5.2.3.2.1

Cancel the common factor.

$d = \frac{1}{2 \cdot 1}$

Step 5.2.3.2.2

Rewrite the expression.

$d = \frac{1}{2}$

Step 5.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{1^{2}}{4 \cdot 1}$

Step 5.2.4.2

Simplify the right side.

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Step 5.2.4.2.1

Simplify each term.

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Step 5.2.4.2.1.1

One to any power is one.

$e = 0 - \frac{1}{4 \cdot 1}$

Step 5.2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{1}{4}$

Step 5.2.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

$e = - \frac{1}{4}$

Step 5.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{1}{2})}^{2} - \frac{1}{4}$ .

${(x + \frac{1}{2})}^{2} - \frac{1}{4}$

Step 5.3

Substitute ${(x + \frac{1}{2})}^{2} - \frac{1}{4}$ for $x^{2} + x$ in the equation $x^{2} + y^{2} + x - y = 0$ .

${(x + \frac{1}{2})}^{2} - \frac{1}{4} + y^{2} - y = 0$

Step 5.4

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

${(x + \frac{1}{2})}^{2} + y^{2} - y = 0 + \frac{1}{4}$

Step 5.5

Complete the square for $y^{2} - y$ .

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Step 5.5.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 1$

$c = 0$

Step 5.5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 1}{2 \cdot 1}$

Step 5.5.3.2

Simplify the right side.

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Step 5.5.3.2.1

Cancel the common factor of $- 1$ and $1$ .

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Step 5.5.3.2.1.1

Rewrite $- 1$ as $- 1 (1)$ .

$d = \frac{- 1 (1)}{2 \cdot 1}$

Step 5.5.3.2.1.2

Cancel the common factor.

$d = \frac{- 1 \cdot 1}{2 \cdot 1}$

Step 5.5.3.2.1.3

Rewrite the expression.

$d = \frac{- 1}{2}$

Step 5.5.3.2.2

Move the negative in front of the fraction.

$d = - \frac{1}{2}$

Step 5.5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 1)}^{2}}{4 \cdot 1}$

Step 5.5.4.2

Simplify the right side.

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Step 5.5.4.2.1

Simplify each term.

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Step 5.5.4.2.1.1

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1}{4 \cdot 1}$

Step 5.5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{1}{4}$

Step 5.5.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

$e = - \frac{1}{4}$

Step 5.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{1}{2})}^{2} - \frac{1}{4}$ .

${(y - \frac{1}{2})}^{2} - \frac{1}{4}$

Step 5.6

Substitute ${(y - \frac{1}{2})}^{2} - \frac{1}{4}$ for $y^{2} - y$ in the equation $x^{2} + y^{2} + x - y = 0$ .

${(x + \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} - \frac{1}{4} = 0 + \frac{1}{4}$

Step 5.7

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

${(x + \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = 0 + \frac{1}{4} + \frac{1}{4}$

Step 5.8

Simplify $0 + \frac{1}{4} + \frac{1}{4}$ .

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Step 5.8.1

Combine the numerators over the common denominator.

${(x + \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{1 + 1}{4}$

Step 5.8.2

Add $1$ and $1$ .

${(x + \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{2}{4}$

Step 5.8.3

Cancel the common factor of $2$ and $4$ .

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Step 5.8.3.1

Factor $2$ out of $2$ .

${(x + \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{2 (1)}{4}$

Step 5.8.3.2

Cancel the common factors.

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Step 5.8.3.2.1

Factor $2$ out of $4$ .

${(x + \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{2 \cdot 1}{2 \cdot 2}$

Step 5.8.3.2.2

Cancel the common factor.

${(x + \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{2 \cdot 1}{2 \cdot 2}$

Step 5.8.3.2.3

Rewrite the expression.

${(x + \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{1}{2}$