Algebra Examples

$\frac{5}{x - 5} = \frac{6}{x}$

Step 1

Subtract $\frac{6}{x}$ from both sides of the equation.

$\frac{5}{x - 5} - \frac{6}{x} = 0$

Step 2

Simplify $\frac{5}{x - 5} - \frac{6}{x}$ .

Tap for more steps...

Step 2.1

To write $\frac{5}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{5}{x - 5} \cdot \frac{x}{x} - \frac{6}{x} = 0$

Step 2.2

To write $- \frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{5}{x - 5} \cdot \frac{x}{x} - \frac{6}{x} \cdot \frac{x - 5}{x - 5} = 0$

Step 2.3

Write each expression with a common denominator of $(x - 5) x$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.3.1

Multiply $\frac{5}{x - 5}$ by $\frac{x}{x}$ .

$\frac{5 x}{(x - 5) x} - \frac{6}{x} \cdot \frac{x - 5}{x - 5} = 0$

Step 2.3.2

Multiply $\frac{6}{x}$ by $\frac{x - 5}{x - 5}$ .

$\frac{5 x}{(x - 5) x} - \frac{6 (x - 5)}{x (x - 5)} = 0$

Step 2.3.3

Reorder the factors of $(x - 5) x$ .

$\frac{5 x}{x (x - 5)} - \frac{6 (x - 5)}{x (x - 5)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 x - 6 (x - 5)}{x (x - 5)} = 0$

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Apply the distributive property.

$\frac{5 x - 6 x - 6 \cdot - 5}{x (x - 5)} = 0$

Step 2.5.2

Multiply $- 6$ by $- 5$ .

$\frac{5 x - 6 x + 30}{x (x - 5)} = 0$

Step 2.5.3

Subtract $6 x$ from $5 x$ .

$\frac{- x + 30}{x (x - 5)} = 0$

Step 2.6

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 30}{x (x - 5)} = 0$

Step 2.7

Rewrite $30$ as $- 1 (- 30)$ .

$\frac{- (x) - 1 (- 30)}{x (x - 5)} = 0$

Step 2.8

Factor $- 1$ out of $- (x) - 1 (- 30)$ .

$\frac{- (x - 30)}{x (x - 5)} = 0$

Step 2.9

Rewrite $- (x - 30)$ as $- 1 (x - 30)$ .

$\frac{- 1 (x - 30)}{x (x - 5)} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{x - 30}{x (x - 5)} = 0$

Step 3

Set the denominator in $\frac{x - 30}{x (x - 5)}$ equal to $0$ to find where the expression is undefined.

$x (x - 5) = 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

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

$x = 0$

$x - 5 = 0$

Step 4.2

Set $x$ equal to $0$ .

$x = 0$

Step 4.3

Set $x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.4

The final solution is all the values that make $x (x - 5) = 0$ true.

$x = 0, 5$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 0, x = 5$

Step 6