Solve the fractional linear equation (10x/3) + (5/2)(2 − x/3) = 7/2 and find the exact value of x.

Introduction / Context:
This equation involves several fractional terms and a bracket, making it a good test of algebraic manipulation. You need to expand correctly, combine like terms, and then solve the resulting linear equation. Such problems are standard in aptitude tests under the topic of simplification or linear equations.

Given Data / Assumptions:

• Equation: (10x/3) + (5/2)(2 − x/3) = 7/2.
• x is a real unknown.
• All denominators (3 and 2) are non zero.
• Standard rules for distribution and fraction arithmetic apply.

Concept / Approach:
The equation can be simplified by first expanding (5/2)(2 − x/3). Then, we add the resulting x terms together and move all terms to one side to isolate x. It can be helpful to clear denominators at some stage by multiplying both sides of the equation by a common multiple of 2 and 3, which is 6. This converts the equation into one with integer coefficients and reduces arithmetic errors.

Step-by-Step Solution:
Start from (10x/3) + (5/2)(2 − x/3) = 7/2. Expand the bracket: (5/2)(2 − x/3) = (5/2)·2 − (5/2)(x/3). This simplifies to 5 − (5x/6). So the equation becomes (10x/3) + 5 − (5x/6) = 7/2. Rewrite 10x/3 with denominator 6: 10x/3 = 20x/6. Thus 20x/6 − 5x/6 + 5 = 7/2. Combine x terms: (20x/6 − 5x/6) = 15x/6 = 5x/2. So we have 5x/2 + 5 = 7/2. Subtract 5 from both sides: 5x/2 = 7/2 − 5. Write 5 as 10/2: 7/2 − 10/2 = −3/2. Thus 5x/2 = −3/2. Multiply both sides by 2: 5x = −3, so x = −3/5.

Verification / Alternative check:
Substitute x = −3/5 into the original equation. Compute 10x/3 = 10(−3/5)/3 = (−6)/3 = −2. Next, 2 − x/3 becomes 2 − (−3/5)/3 = 2 + 3/15 = 2 + 1/5 = 11/5. Then (5/2)(2 − x/3) = (5/2)(11/5) = 11/2. The left side is −2 + 11/2 = −4/2 + 11/2 = 7/2, which equals the right side. This confirms that x = −3/5 is correct.

Why Other Options Are Wrong:
If you substitute 3/5, −5/3 or 5/3 into the equation, the left side does not simplify to 7/2. The option 0 gives a left side equal to 5, not 7/2. Only x = −3/5 satisfies the equation and balances both sides exactly.

Common Pitfalls:
Students may incorrectly expand (5/2)(2 − x/3) or mishandle the conversion of 10x/3 to a fraction with denominator 6. Some might forget to adjust both sides of the equation when clearing denominators or moving terms, causing sign errors. Following each algebraic step carefully and double checking arithmetic with fractions helps prevent such mistakes.

Final Answer:
Therefore, the solution to the equation is x = −3/5.