If 3x + 2y = 12 and x = 2, what is the value of y?

The variable y equals 3 when the given value x = 2 is substituted into the equation 3x + 2y = 12, producing 3(2) + 2y = 12 → 6 + 2y = 12, then solving for y by subtracting 6 to obtain 2y = 6, and finally dividing by 2 to isolate y = 3 as the solution satisfying both the original equation and the constraint.

A. 1

Substituting x = 2 and y = 1 yields 3(2) + 2(1) = 6 + 2 = 8 ≠ 12, falling 4 units short of the required value. The error likely originates from solving 6 + 2y = 12 by subtracting 6 to get 2y = 6 but then dividing by 6 instead of 2 to obtain y = 1 revealing inverse operation confusion during isolation where students divide by the constant term rather than the coefficient. Students might execute 2y = 6 and recognize division is required but incorrectly compute 6 ÷ 6 = 1 instead of 6 ÷ 2 = 3 through divisor misidentification. Another plausible pathway involves solving 3x + 2y = 12 for y generally to get y = (12 - 3x)/2, substituting x = 2 to get y = (12 - 6)/2 = 6/2 = 3 correctly but then mis-transcribing as 1 through digit substitution under time pressure. The 2-unit deficit in the evaluated expression (8 versus 12) corresponds precisely to 2y with y = 1 versus required y = 3 revealing arithmetic error during the critical division step that propagates undetected without verification, a transitional deficiency requiring explicit coefficient identification protocols before isolation operations to prevent divisor misselection that compromises solution accuracy in increasingly complex equation-solving contexts.

B. 2

This value yields 3(2) + 2(2) = 6 + 4 = 10 ≠ 12, falling 2 units short of the target. The error likely originates from solving 6 + 2y = 12 by subtracting 6 to get 2y = 6 but then executing 6 ÷ 3 = 2 instead of 6 ÷ 2 = 3 through divisor misreading (confusing coefficient 2 with constant 6's factor 3). Students might recognize y must be integer and test y = 2 as "reasonable" guess without full computation, or execute mental calculation 6 + 2y = 12 → 2y = 6 → y = 3 correctly but then adjust downward to 2 through unstructured correction influenced by answer choice proximity. Another plausible pathway involves solving the system incorrectly as 3x + 2y = 12 and x + y = 4 (fabricated second equation) to get y = 2. The consistent 2-unit deficit (10 versus 12) equals precisely the coefficient of y revealing off-by-one error in the solution value where y = 2 produces contribution 4 instead of required 6, diagnostically valuable for identifying learners with partial computation accuracy who execute initial steps correctly but introduce arithmetic errors during final isolation a deficiency requiring explicit step recording with intermediate result verification to prevent mental calculation slips that compromise solution validity in constraint satisfaction problems with real-world implications for resource allocation and optimization where precise variable determination affects feasibility and efficiency.

C. 3

This value correctly satisfies both the equation and constraint through systematic substitution and isolation executed with verification. Substitution step: replace x with 2 in 3x + 2y = 12 → 3(2) + 2y = 12 → 6 + 2y = 12. Isolation step 1: subtract 6 from both sides → 2y = 6. Isolation step 2: divide both sides by 2 → y = 3. Verification: substitute x = 2, y = 3 into original equation → 6 + 6 = 12 ✓. Alternative pathway: solve generally for y → 2y = 12 - 3x → y = (12 - 3x)/2; substitute x = 2 → y = (12 - 6)/2 = 6/2 = 3. Graphical interpretation: the line 3x + 2y = 12 intersects vertical line x = 2 at point (2, 3), confirming solution as intersection coordinates. This solution demonstrates comprehensive mastery of equation solving with constraints including substitution protocol execution, inverse operation sequencing with coefficient management, arithmetic precision during division steps, and solution verification through original equation substitution integrated competencies essential for systems of equations, function evaluation, parametric modeling, and real-world constraint satisfaction problems where variables interact through defined relationships requiring sequential solution approaches with precision in arithmetic operations determining feasibility and optimality in resource-constrained environments.

D. 4

Substituting x = 2 and y = 4 yields 3(2) + 2(4) = 6 + 8 = 14 ≠ 12, exceeding the target by 2 units. The error likely originates from solving 6 + 2y = 12 by adding 6 instead of subtracting (6 + 2y + 6 = 12 + 6 → 2y = 18 → y = 9) then adjusting downward to 4 through unstructured approximation, or more plausibly executing 2y = 12 - 6 = 6 correctly but then multiplying by 2/3 instead of dividing by 2 to get y = 4. Students might solve 3x + 2y = 12 for x instead of y: x = (12 - 2y)/3, substitute x = 2 to get 2 = (12 - 2y)/3 → 6 = 12 - 2y → 2y = 6 → y = 3 correctly but then mis-transcribe as 4. Another plausible pathway involves recognizing 3(2) = 6 and 12 - 6 = 6 but then executing 6 ÷ 1.5 = 4 through divisor miscalculation. The 2-unit excess (14 versus 12) equals precisely the coefficient of y revealing off-by-one error in the opposite direction from option B, symmetrically bracketing the correct value and highlighting how arithmetic errors during isolation produce solutions that nearly satisfy the equation but fail verification, requiring explicit verification protocols to catch near-miss solutions that compromise constraint satisfaction accuracy in optimization contexts where precise variable values determine feasibility boundaries and optimal resource allocation.

Conclusion

The solution y = 3 emerges through precise substitution of the constraint value followed by systematic algebraic isolation maintaining equality through balanced operations, with mandatory verification confirming the solution satisfies the original equation. This problem reinforces critical constraint satisfaction competencies essential across mathematical domains: executing substitution as replacement protocol preserving equation structure, applying inverse operations in proper sequence (addressing addition/subtraction before multiplication/division when isolating), managing coefficients during variable isolation with precise division execution, and verifying solutions through original equation substitution to detect arithmetic errors introduced during multi-step manipulation. Mastery of these integrated skills proves indispensable for linear programming (constraint satisfaction), physics (equation of state solutions), economics (market equilibrium determination), and operations research (resource allocation optimization) where accurate variable determination under constraints informs feasible solution identification and optimal decision-making across contexts requiring precise quantitative resolution with real-world implications for efficiency, feasibility, and optimality in resource-constrained environments. The distractors strategically target pervasive arithmetic errors during isolation (divisor misidentification, off-by-one values) and verification omission highlighting the necessity of explicit coefficient identification before division operations and mandatory substitution verification before final answer acceptance to prevent undetected arithmetic slips that compromise solution validity in increasingly complex constraint satisfaction problems with direct implications for optimization accuracy and feasibility determination.