Algebra Examples

$y = \frac{x - 2}{3 x - 1}$

Step 1

Rewrite the equation as $\frac{x - 2}{3 x - 1} = y$ .

$\frac{x - 2}{3 x - 1} = y$

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.

$3 x - 1, 1$

Step 2.2

Remove parentheses.

$3 x - 1, 1$

Step 2.3

The LCM of one and any expression is the expression.

$3 x - 1$

Step 3

Multiply each term in $\frac{x - 2}{3 x - 1} = y$ by $3 x - 1$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{x - 2}{3 x - 1} = y$ by $3 x - 1$ .

$\frac{x - 2}{3 x - 1} (3 x - 1) = y (3 x - 1)$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$\frac{x - 2}{3 x - 1} (3 x - 1) = y (3 x - 1)$

Step 3.2.1.2

Rewrite the expression.

$x - 2 = y (3 x - 1)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify by multiplying through.

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Step 3.3.1.1

Apply the distributive property.

$x - 2 = y (3 x) + y \cdot - 1$

Step 3.3.1.2

Reorder.

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Step 3.3.1.2.1

Rewrite using the commutative property of multiplication.

$x - 2 = 3 y x + y \cdot - 1$

Step 3.3.1.2.2

Move $- 1$ to the left of $y$ .

$x - 2 = 3 y x - 1 \cdot y$

Step 3.3.2

Rewrite $- 1 y$ as $- y$ .

$x - 2 = 3 y x - y$

Step 4

Solve the equation.

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Step 4.1

Subtract $3 y x$ from both sides of the equation.

$x - 2 - 3 y x = - y$

Step 4.2

Add $2$ to both sides of the equation.

$x - 3 y x = - y + 2$

Step 4.3

Factor $x$ out of $x - 3 y x$ .

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Step 4.3.1

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

$x \cdot 1 - 3 y x = - y + 2$

Step 4.3.2

Factor $x$ out of $- 3 y x$ .

$x \cdot 1 + x (- 3 y) = - y + 2$

Step 4.3.3

Factor $x$ out of $x \cdot 1 + x (- 3 y)$ .

$x (1 - 3 y) = - y + 2$

Step 4.4

Divide each term in $x (1 - 3 y) = - y + 2$ by $1 - 3 y$ and simplify.

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Step 4.4.1

Divide each term in $x (1 - 3 y) = - y + 2$ by $1 - 3 y$ .

$\frac{x (1 - 3 y)}{1 - 3 y} = \frac{- y}{1 - 3 y} + \frac{2}{1 - 3 y}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

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

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{x (1 - 3 y)}{1 - 3 y} = \frac{- y}{1 - 3 y} + \frac{2}{1 - 3 y}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y}{1 - 3 y} + \frac{2}{1 - 3 y}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Combine the numerators over the common denominator.

$x = \frac{- y + 2}{1 - 3 y}$

Step 4.4.3.2

Factor $- 1$ out of $- y$ .

$x = \frac{- (y) + 2}{1 - 3 y}$

Step 4.4.3.3

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

$x = \frac{- (y) - 1 (- 2)}{1 - 3 y}$

Step 4.4.3.4

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

$x = \frac{- (y - 2)}{1 - 3 y}$

Step 4.4.3.5

Simplify the expression.

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Step 4.4.3.5.1

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

$x = \frac{- 1 (y - 2)}{1 - 3 y}$

Step 4.4.3.5.2

Move the negative in front of the fraction.

$x = - \frac{y - 2}{1 - 3 y}$