Solve: x/(x−3)=2x/(x−3)−5/3

Verified step by step guidance

1. Start by observing the equation: \(\frac{x}{x-3} = \frac{2x}{x-3} - \frac{5}{3}\). Notice that the denominators on the left and right sides involve \(x-3\), so we should consider the domain restriction \(x \neq 3\) to avoid division by zero.
2. To eliminate the denominators, multiply both sides of the equation by the least common denominator (LCD). The denominators are \(x-3\) and 3, so the LCD is \$3(x-3)\(. Multiply every term by \)3(x-3)$:
3. \[3(x-3) \cdot \frac{x}{x-3} = 3(x-3) \cdot \frac{2x}{x-3} - 3(x-3) \cdot \frac{5}{3}\]
4. Simplify each term by canceling the denominators:
5. \[3x = 3 \cdot 2x - 5(x-3)\]
6. Now, expand and simplify the right side:
7. \[3x = 6x - 5x + 15\]
8. Combine like terms on the right side:
9. \[3x = (6x - 5x) + 15 = x + 15\]
10. Finally, isolate \(x\) by subtracting \(x\) from both sides:
11. \[3x - x = 15\]
12. Simplify the left side:
13. \[2x = 15\]
14. At this point, you can solve for \(x\) by dividing both sides by 2.

Key Concepts

Solving Rational Equations

Rational equations involve expressions with variables in the denominator. To solve them, identify common denominators and eliminate fractions by multiplying both sides accordingly. This simplifies the equation to a polynomial form, making it easier to solve.

Restrictions on the Domain

When solving equations with variables in denominators, certain values can make the denominator zero, which is undefined. Identifying these restrictions is crucial to exclude invalid solutions from the final answer.

Combining Like Terms and Simplifying

After clearing denominators, combine like terms on each side to simplify the equation. This step helps isolate the variable and solve the resulting linear or polynomial equation efficiently.