Trigonometry Examples

$a = \frac{b - 3}{3 b - 6}$

Step 1

Rewrite the equation as $\frac{b - 3}{3 b - 6} = a$ .

$\frac{b - 3}{3 b - 6} = a$

Step 2

Factor $3$ out of $3 b - 6$ .

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Step 2.1

Factor $3$ out of $3 b$ .

$\frac{b - 3}{3 (b) - 6} = a$

Step 2.2

Factor $3$ out of $- 6$ .

$\frac{b - 3}{3 b + 3 \cdot - 2} = a$

Step 2.3

Factor $3$ out of $3 b + 3 \cdot - 2$ .

$\frac{b - 3}{3 (b - 2)} = a$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 (b - 2), 1$

Step 3.2

The LCM of one and any expression is the expression.

$3 (b - 2)$

Step 4

Multiply each term in $\frac{b - 3}{3 (b - 2)} = a$ by $3 (b - 2)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{b - 3}{3 (b - 2)} = a$ by $3 (b - 2)$ .

$\frac{b - 3}{3 (b - 2)} (3 (b - 2)) = a (3 (b - 2))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Rewrite using the commutative property of multiplication.

$3 \frac{b - 3}{3 (b - 2)} (b - 2) = a (3 (b - 2))$

Step 4.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.1

Cancel the common factor.

$3 \frac{b - 3}{3 (b - 2)} (b - 2) = a (3 (b - 2))$

Step 4.2.2.2

Rewrite the expression.

$\frac{b - 3}{b - 2} (b - 2) = a (3 (b - 2))$

Step 4.2.3

Cancel the common factor of $b - 2$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{b - 3}{b - 2} (b - 2) = a (3 (b - 2))$

Step 4.2.3.2

Rewrite the expression.

$b - 3 = a (3 (b - 2))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Rewrite using the commutative property of multiplication.

$b - 3 = 3 a (b - 2)$

Step 4.3.2

Apply the distributive property.

$b - 3 = 3 a b + 3 a \cdot - 2$

Step 4.3.3

Multiply $- 2$ by $3$ .

$b - 3 = 3 a b - 6 a$

Step 5

Solve the equation.

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Step 5.1

Subtract $3 a b$ from both sides of the equation.

$b - 3 - 3 a b = - 6 a$

Step 5.2

Add $3$ to both sides of the equation.

$b - 3 a b = - 6 a + 3$

Step 5.3

Factor $b$ out of $b - 3 a b$ .

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Step 5.3.1

Factor $b$ out of $b^{1}$ .

$b \cdot 1 - 3 a b = - 6 a + 3$

Step 5.3.2

Factor $b$ out of $- 3 a b$ .

$b \cdot 1 + b (- 3 a) = - 6 a + 3$

Step 5.3.3

Factor $b$ out of $b \cdot 1 + b (- 3 a)$ .

$b (1 - 3 a) = - 6 a + 3$

Step 5.4

Divide each term in $b (1 - 3 a) = - 6 a + 3$ by $1 - 3 a$ and simplify.

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Step 5.4.1

Divide each term in $b (1 - 3 a) = - 6 a + 3$ by $1 - 3 a$ .

$\frac{b (1 - 3 a)}{1 - 3 a} = \frac{- 6 a}{1 - 3 a} + \frac{3}{1 - 3 a}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $1 - 3 a$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{b (1 - 3 a)}{1 - 3 a} = \frac{- 6 a}{1 - 3 a} + \frac{3}{1 - 3 a}$

Step 5.4.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 6 a}{1 - 3 a} + \frac{3}{1 - 3 a}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$b = \frac{- 6 a + 3}{1 - 3 a}$

Step 5.4.3.2

Factor $3$ out of $- 6 a + 3$ .

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Step 5.4.3.2.1

Factor $3$ out of $- 6 a$ .

$b = \frac{3 (- 2 a) + 3}{1 - 3 a}$

Step 5.4.3.2.2

Factor $3$ out of $3$ .

$b = \frac{3 (- 2 a) + 3 (1)}{1 - 3 a}$

Step 5.4.3.2.3

Factor $3$ out of $3 (- 2 a) + 3 (1)$ .

$b = \frac{3 (- 2 a + 1)}{1 - 3 a}$

Step 5.4.3.3

Factor $- 1$ out of $- 2 a$ .

$b = \frac{3 (- (2 a) + 1)}{1 - 3 a}$

Step 5.4.3.4

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

$b = \frac{3 (- (2 a) - 1 (- 1))}{1 - 3 a}$

Step 5.4.3.5

Factor $- 1$ out of $- (2 a) - 1 (- 1)$ .

$b = \frac{3 (- (2 a - 1))}{1 - 3 a}$

Step 5.4.3.6

Simplify the expression.

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Step 5.4.3.6.1

Rewrite $- (2 a - 1)$ as $- 1 (2 a - 1)$ .

$b = \frac{3 (- 1 (2 a - 1))}{1 - 3 a}$

Step 5.4.3.6.2

Move the negative in front of the fraction.

$b = - \frac{3 (2 a - 1)}{1 - 3 a}$