Precalculus Examples

$\frac{A t - b}{c - b (1 - 2 t)} = 3$

Step 1

Factor each term.

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Step 1.1

Apply the distributive property.

$\frac{A t - b}{c - b \cdot 1 - b (- 2 t)} = 3$

Step 1.2

Multiply $- 1$ by $1$ .

$\frac{A t - b}{c - b - b (- 2 t)} = 3$

Step 1.3

Rewrite using the commutative property of multiplication.

$\frac{A t - b}{c - b - 1 \cdot - 2 b t} = 3$

Step 1.4

Multiply $- 1$ by $- 2$ .

$\frac{A t - b}{c - b + 2 b t} = 3$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$c - b + 2 b t, 1$

Step 2.2

Remove parentheses.

$c - b + 2 b t, 1$

Step 2.3

The LCM of one and any expression is the expression.

$c - b + 2 b t$

Step 3

Multiply each term in $\frac{A t - b}{c - b + 2 b t} = 3$ by $c - b + 2 b t$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{A t - b}{c - b + 2 b t} = 3$ by $c - b + 2 b t$ .

$\frac{A t - b}{c - b + 2 b t} (c - b + 2 b t) = 3 (c - b + 2 b t)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $c - b + 2 b t$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{A t - b}{c - b + 2 b t} (c - b + 2 b t) = 3 (c - b + 2 b t)$

Step 3.2.1.2

Rewrite the expression.

$A t - b = 3 (c - b + 2 b t)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$A t - b = 3 c + 3 (- b) + 3 (2 b t)$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $- 1$ by $3$ .

$A t - b = 3 c - 3 b + 3 (2 b t)$

Step 3.3.2.2

Multiply $2$ by $3$ .

$A t - b = 3 c - 3 b + 6 (b t)$

$A t - b = 3 c - 3 b + 6 b t$

Step 4

Solve the equation.

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Step 4.1

Subtract $6 b t$ from both sides of the equation.

$A t - b - 6 b t = 3 c - 3 b$

Step 4.2

Move all terms not containing $t$ to the right side of the equation.

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Step 4.2.1

Add $b$ to both sides of the equation.

$A t - 6 b t = 3 c - 3 b + b$

Step 4.2.2

Add $- 3 b$ and $b$ .

$A t - 6 b t = 3 c - 2 b$

Step 4.3

Factor $t$ out of $A t - 6 b t$ .

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Step 4.3.1

Factor $t$ out of $A t$ .

$t A - 6 b t = 3 c - 2 b$

Step 4.3.2

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

$t A + t (- 6 b) = 3 c - 2 b$

Step 4.3.3

Factor $t$ out of $t A + t (- 6 b)$ .

$t (A - 6 b) = 3 c - 2 b$

Step 4.4

Divide each term in $t (A - 6 b) = 3 c - 2 b$ by $A - 6 b$ and simplify.

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Step 4.4.1

Divide each term in $t (A - 6 b) = 3 c - 2 b$ by $A - 6 b$ .

$\frac{t (A - 6 b)}{A - 6 b} = \frac{3 c}{A - 6 b} + \frac{- 2 b}{A - 6 b}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $A - 6 b$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{t (A - 6 b)}{A - 6 b} = \frac{3 c}{A - 6 b} + \frac{- 2 b}{A - 6 b}$

Step 4.4.2.1.2

Divide $t$ by $1$ .

$t = \frac{3 c}{A - 6 b} + \frac{- 2 b}{A - 6 b}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Combine the numerators over the common denominator.

$t = \frac{3 c - 2 b}{A - 6 b}$