Find the Holes in the Graph y=((x-3)(x-1))/((x-1)(x-5))

$y = \frac{(x - 3) (x - 1)}{(x - 1) (x - 5)}$

Step 1

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$y = \frac{(x - 3) (x - 1)}{(x - 1) (x - 5)}$

Rewrite the expression.

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

Step 2

To find the holes in the graph, look at the denominator factors that were cancelled.

$x - 1$

Step 3

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{x - 3}{x - 5}$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Substitute $1$ for $x$ in $\frac{x - 3}{x - 5}$ and simplify.

Substitute $1$ for $x$ to find the $y$ coordinate of the hole.

$\frac{1 - 3}{1 - 5}$

Simplify.

Subtract $3$ from $1$ .

$\frac{- 2}{1 - 5}$

Subtract $5$ from $1$ .

$\frac{- 2}{- 4}$

Cancel the common factor of $- 2$ and $- 4$ .

Factor $- 2$ out of $- 2$ .

$\frac{- 2 (1)}{- 4}$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

$\frac{- 2 \cdot 1}{- 2 \cdot 2}$

Cancel the common factor.

$\frac{- 2 \cdot 1}{- 2 \cdot 2}$

Rewrite the expression.

$\frac{1}{2}$

The holes in the graph are the points where any of the cancelled factors are equal to $0$ .

$(1, \frac{1}{2})$

Step 4