What is the value of b in this equation? 3b - 5 = 16

The value of b equals 7 in the equation 3b - 5 = 16 when algebraic isolation procedures are executed by first adding 5 to both sides to eliminate the constant term, yielding 3b = 21, then dividing both sides by 3 to isolate the variable, producing b = 7 as the solution that satisfies the original equation upon verification.

A. 13

Substituting b = 13 into the original equation yields 3(13) - 5 = 39 - 5 = 34, which substantially exceeds the target value of 16 by 18 units. This error likely originates from misapplying inverse operations specifically adding 5 to 16 to get 21, then incorrectly adding 3 rather than dividing by 3 (21 + 3 = 24, then perhaps subtracting 11 through unstructured adjustment to reach 13). Alternatively, students might confuse the equation with 3b + 5 = 16, solving to get 3b = 11 then approximating b ≈ 3.67 and erroneously rounding upward to 13 through digit transposition. The magnitude discrepancy (34 versus 16) provides substantial separation from the correct result, making this option effective at identifying fundamental procedural errors in equation solving rather than minor computational slips.

B. 24

This value produces 3(24) - 5 = 72 - 5 = 67 when substituted more than quadruple the target value of 16. The error pattern likely involves compounding multiple procedural mistakes: first adding 5 to 16 correctly (yielding 21), then mistakenly multiplying by 3 instead of dividing (21 × 3 = 63), followed by arbitrary adjustment to 24 through unclear reasoning. Another plausible pathway involves misreading the coefficient as division rather than multiplication (b ÷ 3 - 5 = 16), solving to get b ÷ 3 = 21 then b = 63, then erroneously dividing by 2.625 to reach 24. The extreme deviation from the correct answer makes this option particularly useful for diagnosing severe conceptual misunderstandings about inverse operations or coefficient interpretation in linear equations.

C. 7

This value correctly satisfies the equation through systematic algebraic manipulation executed in two verification pathways. Primary solution method: add 5 to both sides (3b - 5 + 5 = 16 + 5 → 3b = 21), then divide both sides by 3 (3b ÷ 3 = 21 ÷ 3 → b = 7). Verification substitution: 3(7) - 5 = 21 - 5 = 16, confirming exact equality with the original equation's right-hand side. This solution demonstrates complete mastery of fundamental equation-solving principles: recognizing that operations must be applied equally to both sides to maintain equality, executing inverse operations in reverse order of operations (addressing subtraction before multiplication), and verifying solutions through substitution to guard against procedural errors. The integer solution reflects appropriate problem design where coefficients and constants yield whole-number results, reinforcing student confidence in algebraic manipulation while avoiding unnecessary fractional complexity that might obscure core procedural understanding.

D. 21

Substituting b = 21 yields 3(21) - 5 = 63 - 5 = 58 more than triple the target value. This error represents a classic "stopped-short" mistake where students correctly execute the first algebraic step (adding 5 to both sides to get 3b = 21) but then fail to complete the isolation process by dividing by 3, instead incorrectly treating the intermediate result 21 as the final answer for b. This pattern reveals incomplete understanding of variable isolation requirements specifically that coefficients must be eliminated through division (for multiplication) or multiplication (for division) to achieve standalone variable representation. The proximity of 21 to the intermediate calculation step makes this an exceptionally effective distractor targeting students with partial procedural knowledge who recognize initial steps but lack comprehensive solution sequencing awareness.

Conclusion

The solution b = 7 emerges through rigorous application of algebraic isolation techniques following the principle that equations maintain equality only when identical operations are applied to both sides. This problem exemplifies foundational linear equation solving essential for advanced algebraic manipulation, function analysis, and real-world modeling where unknown quantities must be systematically determined. Mastery demands understanding inverse operation sequencing (addressing addition/subtraction before multiplication/division when isolating variables), maintaining equality through balanced transformations, and verifying solutions to prevent undetected procedural errors. The "stopped-short" error represented by option D proves particularly instructive, highlighting how partial procedural knowledge can produce intermediate values mistaken for final solutions a misconception requiring explicit instructional attention to develop complete solution pathway awareness. Students developing automaticity with these structured approaches build robust algebraic reasoning capabilities transferable across mathematical domains where equation solving forms the backbone of quantitative problem resolution.