Solve: x/(x−3)=2x/(x−3)−5/3

Verified step by step guidance

1. Start by observing that the equation \( \frac{x}{x-3} = \frac{2x}{x-3} - \frac{5}{3} \) has a common denominator on the left and part of the right side.
2. To eliminate the fractions, multiply every term by the least common denominator, which is \((x-3)\). This will give you: \( x = 2x - \frac{5}{3}(x-3) \).
3. Distribute the \(-\frac{5}{3}\) across \((x-3)\) to simplify the equation: \( x = 2x - \frac{5}{3}x + 5 \).
4. Combine like terms on the right side of the equation: \( x = (2 - \frac{5}{3})x + 5 \).
5. Isolate \(x\) by moving all terms involving \(x\) to one side and constant terms to the other, then solve for \(x\).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions is crucial for solving equations involving them. In this problem, both sides of the equation contain a common denominator, which allows for simplification and elimination of the denominator when solving for x.

Cross Multiplication

Cross multiplication is a technique used to eliminate fractions in equations. When two fractions are set equal to each other, you can multiply the numerator of one fraction by the denominator of the other. This method simplifies the equation, making it easier to isolate the variable and solve for its value.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This process often includes combining like terms, isolating the variable, and performing inverse operations. In this case, after simplifying the equation, you will need to rearrange and solve for x to find the solution.