Difficulty: Medium
Correct Answer: -1
Explanation:
Introduction / Context:
This is another fractional linear equation often seen in aptitude tests. It requires expansion of a bracket, careful handling of fractions, and rearrangement to isolate x. The aim is to solve without arithmetic mistakes while simplifying the denominators.
Given Data / Assumptions:
Concept / Approach:
We expand the bracket inside the numerator, divide by 2, and then combine the result with 17/3. To remove fractional denominators, we can multiply through by the least common multiple of 3 and 2 which is 6, converting the equation into one with integer coefficients.
Step-by-Step Solution:
Start from 17/3 + [3(2x − 5/3)]/2 = 1/6
Expand inside bracket: 3(2x − 5/3) = 6x − 5
So the equation becomes 17/3 + (6x − 5)/2 = 1/6
Multiply the entire equation by 6 to clear denominators
6 * (17/3) + 6 * (6x − 5)/2 = 6 * (1/6)
This gives 34 + 3(6x − 5) = 1
34 + 18x − 15 = 1
Combine constants: 18x + 19 = 1
18x = 1 − 19 = −18
x = −18 / 18 = −1
Verification / Alternative check:
Substitute x = −1 into the original equation. Then 2x − 5/3 becomes −2 − 5/3 = −11/3. Multiply by 3 to get −11, then divide by 2 to get −11/2. So left side is 17/3 − 11/2. Converting to a common denominator 6, this equals 34/6 − 33/6 = 1/6, which equals the right side, confirming correctness.
Why Other Options Are Wrong:
Values 1, 3 and −3 do not yield 1/6 when substituted and evaluated. Zero would leave the bracket with a negative value that does not balance the equation. Only −1 makes both sides equal to 1/6, so the rest must be rejected.
Common Pitfalls:
Common issues include distributing 3 incorrectly inside the bracket, forgetting to multiply every term by 6, or mishandling the sign when subtracting 5. Another trap is combining the constant terms incorrectly, which leads to a wrong linear equation in x.
Final Answer:
Therefore, the solution to the equation is x = −1.
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