If 4x - 3y = 12 and y = 4, what is the value of x?

The variable x equals 6 when the given value y = 4 is substituted into the equation 4x - 3y = 12, producing 4x - 12 = 12 after evaluating -3(4), then solving for x by adding 12 to both sides to obtain 4x = 24, and finally dividing by 4 to isolate x = 6 as the solution satisfying both the original equation and the constraint.

A. 3

Substituting x = 3 and y = 4 yields 4(3) - 3(4) = 12 - 12 = 0, which falls 12 units short of the required 12 on the right-hand side. This error likely originates from solving 4x = 12 directly without accounting for the -3y term effectively ignoring the y-contribution entirely. Students might execute 4x = 12 → x = 3 after seeing the constant 12 on the right, demonstrating failure to incorporate all equation components during solution. Another plausible pathway involves correctly substituting y = 4 to get 4x - 12 = 12 but then erroneously subtracting 12 from both sides (4x = 0) rather than adding, yielding x = 0 then mis-transcribing as 3. The complete elimination of the left side's value (resulting in 0) reveals a fundamental procedural error in handling negative terms during equation manipulation a critical deficiency as sign management errors produce solutions that satisfy modified rather than original equations, with potentially severe consequences in applied contexts like engineering equilibrium calculations or financial balance equations where term omission invalidates entire solution frameworks.

B. 6

This value correctly satisfies both the equation and constraint through systematic substitution and isolation executed with verification. Substitution step: replace y with 4 in 4x - 3y = 12 → 4x - 3(4) = 12 → 4x - 12 = 12. Isolation step 1: add 12 to both sides → 4x - 12 + 12 = 12 + 12 → 4x = 24. Isolation step 2: divide both sides by 4 → 4x ÷ 4 = 24 ÷ 4 → x = 6. Verification: substitute x = 6, y = 4 into original equation → 4(6) - 3(4) = 24 - 12 = 12, confirming equality. Alternative verification pathway: solve generally for x in terms of y → 4x = 12 + 3y → x = (12 + 3y)/4; substitute y = 4 → x = (12 + 12)/4 = 24/4 = 6. This solution demonstrates comprehensive mastery of equation solving with constraints including substitution protocol execution, sign management during term transposition, inverse operation sequencing, and solution verification 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.

C. 9

Substituting x = 9 and y = 4 yields 4(9) - 3(4) = 36 - 12 = 24, which exceeds the target value by 12 units precisely double the required result. This error likely originates from solving 4x = 36 (perhaps by adding 12 + 12 + 12 instead of 12 + 12) then dividing by 4 to get 9. Students might execute 4x - 12 = 12 correctly but then add 24 instead of 12 to both sides (4x = 36) through arithmetic error or misreading the constant. Another plausible pathway involves confusing the equation with 4x - 3y = 24 (double the actual constant) then solving correctly for that modified equation. The consistent doubling pattern (24 versus 12) suggests possible misinterpretation of coefficients or constants through visual scanning errors particularly relevant as the numbers 12 and 24 appear in close proximity during solution steps, creating cognitive load that may trigger transcription errors under time pressure. This distractor effectively identifies students who execute algebraic manipulation correctly but introduce arithmetic errors during constant combination a transitional deficiency requiring emphasis on verification protocols and careful transcription practices during multi-step calculations where intermediate values create opportunities for digit substitution or magnitude misjudgment.

D. 12

This value produces 4(12) - 3(4) = 48 - 12 = 36 upon substitution triple the target value of 12. The error likely originates from solving 4x = 48 (perhaps by multiplying 12 × 4 instead of adding 12 + 12) then dividing by 4 to get x = 12. Students might execute the substitution step correctly (4x - 12 = 12) but then mistakenly multiply both sides by 4 rather than adding 12 (4x = 48) through inverse operation confusion treating the -12 as requiring multiplication rather than addition for elimination. Another plausible pathway involves misreading the original equation as 4x ÷ 3y = 12 or other structural misinterpretation leading to erroneous solution pathway. The tripling pattern (36 versus 12) reveals systematic coefficient misapplication rather than random error diagnostically valuable for identifying students who confuse operation types during equation manipulation (multiplication versus addition for term elimination), a fundamental misconception requiring explicit instruction on inverse operation selection based on term connectivity (addition/subtraction versus multiplication/division) within equation structures.

Conclusion

The solution x = 6 emerges through precise substitution of the constraint value followed by systematic algebraic isolation maintaining equality through balanced operations. This problem exemplifies foundational constraint satisfaction techniques essential for systems of equations, function evaluation, and parametric problem-solving where variables interact through defined relationships requiring sequential solution approaches. Mastery demands understanding substitution as replacement protocol preserving equation structure, executing inverse operations in proper sequence (addressing addition/subtraction before multiplication/division when isolating), managing signs during term transposition with precision, and verifying solutions through original equation substitution to detect procedural errors. The arithmetic error patterns in options C and D highlight a critical transitional challenge: maintaining computational accuracy during multi-step algebraic manipulation where intermediate values create cognitive load that may trigger transcription errors or operation confusion deficiencies requiring explicit development of verification habits and careful transcription practices. Students developing automaticity with these integrated approaches build robust algebraic reasoning capabilities transferable to linear systems, optimization problems, and real-world modeling where constraint satisfaction forms the backbone of quantitative decision-making across engineering, economics, and operations research domains.