Algebra Examples

$s (y) = \frac{3 y}{y + 5}$

Step 1

Rewrite the equation as $\frac{3 y}{y + 5} = x$ .

$\frac{3 y}{y + 5} = x$

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.

$y + 5, 1$

Step 2.2

Remove parentheses.

$y + 5, 1$

Step 2.3

The LCM of one and any expression is the expression.

$y + 5$

Step 3

Multiply each term in $\frac{3 y}{y + 5} = x$ by $y + 5$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{3 y}{y + 5} = x$ by $y + 5$ .

$\frac{3 y}{y + 5} (y + 5) = x (y + 5)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $y + 5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 y}{y + 5} (y + 5) = x (y + 5)$

Step 3.2.1.2

Rewrite the expression.

$3 y = x (y + 5)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$3 y = x y + x \cdot 5$

Step 3.3.2

Move $5$ to the left of $x$ .

$3 y = x y + 5 x$

Step 4

Solve the equation.

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Step 4.1

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

$3 y - x y = 5 x$

Step 4.2

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

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Step 4.2.1

Factor $y$ out of $3 y$ .

$y \cdot 3 - x y = 5 x$

Step 4.2.2

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

$y \cdot 3 + y (- x) = 5 x$

Step 4.2.3

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

$y (3 - x) = 5 x$

Step 4.3

Divide each term in $y (3 - x) = 5 x$ by $3 - x$ and simplify.

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Step 4.3.1

Divide each term in $y (3 - x) = 5 x$ by $3 - x$ .

$\frac{y (3 - x)}{3 - x} = \frac{5 x}{3 - x}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $3 - x$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{y (3 - x)}{3 - x} = \frac{5 x}{3 - x}$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x}{3 - x}$

Step 5

Set the denominator in $\frac{5 x}{3 - x}$ equal to $0$ to find where the expression is undefined.

$3 - x = 0$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 6.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- x = - 3$ by $- 1$ .

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2.2

Divide $x$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 7

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x \neq 3}$

Step 8

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - 5) \cup (- 5, \infty)$

Set-Builder Notation:

${y | y \neq - 5}$

Step 9

Determine the domain and range.

Domain: $(- \infty, 3) \cup (3, \infty), {x | x \neq 3}$

Range: $(- \infty, - 5) \cup (- 5, \infty), {y | y \neq - 5}$

Step 10