Precalculus Examples

$\frac{y^{2} - 49}{{(y - 7)}^{2}} \div \frac{5 y + 35}{y^{2} - 7 y}$

Step 1

Set the denominator in $\frac{y^{2} - 49}{{(y - 7)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(y - 7)}^{2} = 0$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Set the $y - 7$ equal to $0$ .

$y - 7 = 0$

Step 2.2

Add $7$ to both sides of the equation.

$y = 7$

Step 3

Set the denominator in $\frac{5 y + 35}{y^{2} - 7 y}$ equal to $0$ to find where the expression is undefined.

$y^{2} - 7 y = 0$

Step 4

Solve for $y$ .

Tap for more steps...

Step 4.1

Factor $y$ out of $y^{2} - 7 y$ .

Tap for more steps...

Step 4.1.1

Factor $y$ out of $y^{2}$ .

$y \cdot y - 7 y = 0$

Step 4.1.2

Factor $y$ out of $- 7 y$ .

$y \cdot y + y \cdot - 7 = 0$

Step 4.1.3

Factor $y$ out of $y \cdot y + y \cdot - 7$ .

$y (y - 7) = 0$

Step 4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$y - 7 = 0$

Step 4.3

Set $y$ equal to $0$ .

$y = 0$

Step 4.4

Set $y - 7$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 4.4.1

Set $y - 7$ equal to $0$ .

$y - 7 = 0$

Step 4.4.2

Add $7$ to both sides of the equation.

$y = 7$

Step 4.5

The final solution is all the values that make $y (y - 7) = 0$ true.

$y = 0, 7$

Step 5

Set the denominator in $\frac{y^{2} - 49}{{(y - 7)}^{2}} \div \frac{5 y + 35}{y^{2} - 7 y}$ equal to $0$ to find where the expression is undefined.

$\frac{5 y + 35}{y^{2} - 7 y} = 0$

Step 6

Solve for $y$ .

Tap for more steps...

Step 6.1

Set the numerator equal to zero.

$5 y + 35 = 0$

Step 6.2

Solve the equation for $y$ .

Tap for more steps...

Step 6.2.1

Subtract $35$ from both sides of the equation.

$5 y = - 35$

Step 6.2.2

Divide each term in $5 y = - 35$ by $5$ and simplify.

Tap for more steps...

Step 6.2.2.1

Divide each term in $5 y = - 35$ by $5$ .

$\frac{5 y}{5} = \frac{- 35}{5}$

Step 6.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 35}{5}$

Step 6.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 35}{5}$

Step 6.2.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.2.3.1

Divide $- 35$ by $5$ .

$y = - 7$

Step 7

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

Interval Notation:

$(- \infty, - 7) \cup (- 7, 0) \cup (0, 7) \cup (7, \infty)$

Set-Builder Notation:

${y | y \neq 0, 7, - 7}$

Step 8