Algebra Examples

$x - \frac{7}{x} = 6$

Step 1

Subtract $6$ from both sides of the equation.

$x - \frac{7}{x} - 6 = 0$

Step 2

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

$\frac{x \cdot x}{x} - \frac{7}{x} - 6 = 0$

Step 3

Combine the numerators over the common denominator.

$\frac{x \cdot x - 7}{x} - 6 = 0$

Step 4

Multiply $x$ by $x$ .

$\frac{x^{2} - 7}{x} - 6 = 0$

Step 5

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

$\frac{x^{2} - 7}{x} - 6 \cdot \frac{x}{x} = 0$

Step 6

Combine $- 6$ and $\frac{x}{x}$ .

$\frac{x^{2} - 7}{x} + \frac{- 6 x}{x} = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{x^{2} - 7 - 6 x}{x} = 0$

Step 8

Factor $x^{2} - 7 - 6 x$ using the AC method.

Tap for more steps...

Step 8.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 8.2

Write the factored form using these integers.

$\frac{(x - 7) (x + 1)}{x} = 0$

Step 9

Set the numerator equal to zero.

$(x - 7) (x + 1) = 0$

Step 10

Solve the equation for $x$ .

Tap for more steps...

Step 10.1

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

$x - 7 = 0$

$x + 1 = 0$

Step 10.2

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

Tap for more steps...

Step 10.2.1

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

$x - 7 = 0$

Step 10.2.2

Add $7$ to both sides of the equation.

$x = 7$

Step 10.3

Set $x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 10.3.1

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

$x + 1 = 0$

Step 10.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 10.4

The final solution is all the values that make $(x - 7) (x + 1) = 0$ true.

$x = 7, - 1$