Solve a simple linear equation: find the value of x for which the two linear expressions 5x + 17 and 17x − 5 are equal to each other.

Introduction / Context:
This is a straightforward algebra question involving a linear equation in one variable. You are given two linear expressions, 5x + 17 and 17x − 5, and asked to find the value of x that makes them equal. Such problems are among the most basic in algebra and are often used early in aptitude and entrance exams to test comfort with solving simple equations.

Given Data / Assumptions:

• The two expressions are 5x + 17 and 17x − 5.
• We are told they are equal for a particular value of x.
• We must find that value of x.
• x is a real number.

Concept / Approach:
When two expressions are equal, we can set them equal and solve the resulting equation. Since both sides are linear in x, the equation will have a single solution. The key steps are to collect like terms and isolate x on one side of the equation using basic algebraic operations such as addition, subtraction, and division by a non zero constant.

Step-by-Step Solution:
Step 1: Set the expressions equal: 5x + 17 = 17x − 5.Step 2: Bring all x terms to one side by subtracting 5x from both sides: 17x − 5x − 5 = 17.Step 3: Simplify the x terms: 12x − 5 = 17.Step 4: Add 5 to both sides to isolate the term in x: 12x = 22.Step 5: Divide both sides by 12: x = 22/12.Step 6: Simplify the fraction by dividing numerator and denominator by 2: x = 11/6.

Verification / Alternative check:
Substitute x = 11/6 back into both expressions. For 5x + 17, we obtain 5 × 11/6 + 17 = 55/6 + 102/6 = 157/6. For 17x − 5, we have 17 × 11/6 − 5 = 187/6 − 30/6 = 157/6. Since both expressions give the same value when x = 11/6, this verifies that our solution is correct.

Why Other Options Are Wrong:
The values −11/6 and ±6/11 are obtained by mishandling signs or inverting the fraction improperly during solving. For example, taking 12x = 22 and solving as x = 12/22 gives 6/11, which is incorrect. The option 1 does not satisfy the original equation, since 5(1) + 17 = 22 and 17(1) − 5 = 12. Only x = 11/6 makes both sides equal.

Common Pitfalls:
Typical errors include forgetting to subtract 5x from both sides, misplacing signs when moving terms, or not simplifying the final fraction to lowest terms. Writing down intermediate steps clearly and checking each arithmetic operation helps avoid these mistakes. Always verify the solution by substituting it back into the original expressions.

Final Answer:
The value of x that makes 5x + 17 equal to 17x − 5 is 11/6.