Algebra Examples

$\frac{x}{x - 1} + \frac{x}{3} = \frac{5}{x - 1}$

Step 1

Subtract $\frac{5}{x - 1}$ from both sides of the equation.

$\frac{x}{x - 1} + \frac{x}{3} - \frac{5}{x - 1} = 0$

Step 2

Simplify $\frac{x}{x - 1} + \frac{x}{3} - \frac{5}{x - 1}$ .

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Step 2.1

To write $\frac{x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{x - 1} \cdot \frac{3}{3} + \frac{x}{3} - \frac{5}{x - 1} = 0$

Step 2.2

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{x}{x - 1} \cdot \frac{3}{3} + \frac{x}{3} \cdot \frac{x - 1}{x - 1} - \frac{5}{x - 1} = 0$

Step 2.3

Write each expression with a common denominator of $(x - 1) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{x}{x - 1}$ by $\frac{3}{3}$ .

$\frac{x \cdot 3}{(x - 1) \cdot 3} + \frac{x}{3} \cdot \frac{x - 1}{x - 1} - \frac{5}{x - 1} = 0$

Step 2.3.2

Multiply $\frac{x}{3}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x \cdot 3}{(x - 1) \cdot 3} + \frac{x (x - 1)}{3 (x - 1)} - \frac{5}{x - 1} = 0$

Step 2.3.3

Reorder the factors of $(x - 1) \cdot 3$ .

$\frac{x \cdot 3}{3 (x - 1)} + \frac{x (x - 1)}{3 (x - 1)} - \frac{5}{x - 1} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 3 + x (x - 1)}{3 (x - 1)} - \frac{5}{x - 1} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $x$ out of $x \cdot 3 + x (x - 1)$ .

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Step 2.5.1.1

Factor $x$ out of $x \cdot 3$ .

$\frac{x (3) + x (x - 1)}{3 (x - 1)} - \frac{5}{x - 1} = 0$

Step 2.5.1.2

Factor $x$ out of $x (3) + x (x - 1)$ .

$\frac{x (3 + x - 1)}{3 (x - 1)} - \frac{5}{x - 1} = 0$

Step 2.5.2

Subtract $1$ from $3$ .

$\frac{x (x + 2)}{3 (x - 1)} - \frac{5}{x - 1} = 0$

Step 2.6

To write $- \frac{5}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x (x + 2)}{3 (x - 1)} - \frac{5}{x - 1} \cdot \frac{3}{3} = 0$

Step 2.7

Write each expression with a common denominator of $3 (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

Multiply $\frac{5}{x - 1}$ by $\frac{3}{3}$ .

$\frac{x (x + 2)}{3 (x - 1)} - \frac{5 \cdot 3}{(x - 1) \cdot 3} = 0$

Step 2.7.2

Reorder the factors of $(x - 1) \cdot 3$ .

$\frac{x (x + 2)}{3 (x - 1)} - \frac{5 \cdot 3}{3 (x - 1)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{x (x + 2) - 5 \cdot 3}{3 (x - 1)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - 5 \cdot 3}{3 (x - 1)} = 0$

Step 2.9.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 2 - 5 \cdot 3}{3 (x - 1)} = 0$

Step 2.9.3

Move $2$ to the left of $x$ .

$\frac{x^{2} + 2 \cdot x - 5 \cdot 3}{3 (x - 1)} = 0$

Step 2.9.4

Multiply $- 5$ by $3$ .

$\frac{x^{2} + 2 x - 15}{3 (x - 1)} = 0$

Step 2.9.5

Factor $x^{2} + 2 x - 15$ using the AC method.

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Step 2.9.5.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 2.9.5.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 5)}{3 (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 3) (x + 5) = 0$

Step 4

Solve the equation for $x$ .

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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 $0$ .

$x - 3 = 0$

$x + 5 = 0$

Step 4.2

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.3

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

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Step 4.3.1

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

$x + 5 = 0$

Step 4.3.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4.4

The final solution is all the values that make $(x - 3) (x + 5) = 0$ true.

$x = 3, - 5$