What is the solution to the system of equations: 2x + y = 10 and x - y = 2?

The ordered pair (4, 2) satisfies both equations simultaneously when substituted into 2x + y = 10 yielding 2(4) + 2 = 10 and into x - y = 2 yielding 4 - 2 = 2, confirming it as the unique intersection point of the two linear equations representing the system's solution.

A. (2, 6)

Substituting x = 2 and y = 6 into the first equation yields 2(2) + 6 = 10, which satisfies that equation correctly, but substituting into the second equation produces 2 - 6 = -4 ≠ 2, violating the second constraint. This partial satisfaction reveals students likely solved only the first equation for y in terms of x (y = 10 - 2x) and selected a point on that line without verifying against the second equation a critical procedural omission where learners recognize solution requires satisfying both equations but neglect mandatory dual verification. Students might have graphed only the first line mentally and selected an aesthetically pleasing integer point without cross-checking. The dramatic failure on the second equation (-4 versus 2) with opposite sign indicates complete disregard for the second constraint's requirements, highlighting how single-equation satisfaction provides false confidence without systematic verification protocols. This distractor effectively identifies learners who understand solution concepts theoretically but lack disciplined verification habits essential for systems solving where partial satisfaction produces mathematically invalid solutions with potentially severe consequences in optimization contexts requiring simultaneous constraint satisfaction.

B. (3, 4)

This ordered pair yields 2(3) + 4 = 10 for the first equation (satisfied) but 3 - 4 = -1 ≠ 2 for the second equation (violated), producing a 3-unit deficit from the required difference. The error likely originates from solving the system through addition method but miscalculating the sum: adding equations yields 3x = 12 → x = 4 correctly, but students might have executed 2x + x = 3x and 10 + 2 = 12 correctly yet divided 12 ÷ 4 = 3 instead of 12 ÷ 3 = 4 through divisor misreading. Another plausible pathway involves substitution method where y = x - 2 is substituted into first equation yielding 2x + (x - 2) = 10 → 3x = 12 → x = 4, but students solved 3x = 9 → x = 3 through arithmetic error then computed y = 3 - 2 = 1 but mis-transcribed as y = 4. The consistent x = 3 value across potential error pathways reveals systematic miscalculation during the critical division step where 12 ÷ 3 becomes 12 ÷ 4 a single-digit transposition with cascading consequences producing solutions that appear plausible (integer coordinates) yet fail verification. This distractor proves particularly instructive for diagnosing arithmetic precision deficiencies during algebraic manipulation where intermediate calculation errors propagate undetected without verification.

C. (4, 2)

This ordered pair correctly satisfies both equations through multiple solution pathways with comprehensive verification. Addition method: 2x + y = 10 plus x - y = 2 yields 3x = 12 → x = 4; substitute into second equation: 4 - y = 2 → y = 2. Substitution method: from second equation y = x - 2; substitute into first: 2x + (x - 2) = 10 → 3x = 12 → x = 4; y = 4 - 2 = 2. Graphical verification: plotting both lines reveals intersection at (4, 2) in the first quadrant. Matrix method: coefficient matrix determinant = (2)(-1) - (1)(1) = -3; Cramer's rule yields x = det([-10,1;-2,-1])/-3 = (-10+2)/-3 = 8/3? Wait, correct application: x = det([10,1;2,-1])/-3 = (-10-2)/-3 = 12/3 = 4; y = det([2,10;1,2])/-3 = (4-10)/-3 = 6/3 = 2. Verification substitution: 2(4)+2=10 ✓ and 4-2=2 ✓. This solution demonstrates comprehensive mastery of linear systems including multiple solution methodologies (addition, substitution, graphical, matrix), arithmetic precision during critical operations, and mandatory dual-equation verification ensuring simultaneous satisfaction integrated competencies essential for optimization (linear programming), physics (force equilibrium), economics (supply-demand intersection), and engineering (circuit analysis) where accurate system solution determines equilibrium states and constraint satisfaction across multidimensional problem spaces requiring coordinated variable determination.

D. (5, 0)

Substituting x = 5 and y = 0 yields 2(5) + 0 = 10 for the first equation (satisfied) but 5 - 0 = 5 ≠ 2 for the second equation (violated by 3 units). The error likely originates from solving only the first equation with y = 0 assumption (x-intercept of first line) without considering the second constraint. Students might have recognized 2x = 10 → x = 5 when y = 0 but failed to recognize this represents only one line's intercept, not the system solution. Another plausible pathway involves misreading the second equation as x + y = 2 (sign error) then solving 2x + y = 10 and x + y = 2 to get x = 8, y = -6, then adjusting to (5, 0) through unstructured approximation. The value (5, 0) also represents the x-intercept of the first equation suggesting students confused single-equation features with system solutions, a conceptual deficiency where learners treat individual equation properties as system properties without recognizing solution requires intersection rather than individual characteristics. This distractor effectively identifies students who lack geometric interpretation of systems as line intersections, instead focusing on isolated equation features a foundational gap requiring explicit graphical reinforcement showing how two lines intersect at precisely one point representing the unique solution satisfying both constraints simultaneously.

Conclusion

The solution (4, 2) emerges through rigorous application of system-solving methodologies particularly the addition method that efficiently eliminates y by summing equations followed by back-substitution and mandatory verification in both original equations to ensure simultaneous satisfaction. This problem reinforces critical algebraic competencies essential across quantitative domains: recognizing that system solutions require satisfying all equations concurrently rather than individually, selecting efficient solution methods based on coefficient patterns (addition when coefficients are opposites), executing arithmetic operations with precision during critical steps, and implementing verification protocols that catch calculation errors before final answer submission. Mastery of these integrated skills proves indispensable for operations research (resource allocation optimization), physics (vector component resolution), economics (market equilibrium modeling), and computer graphics (intersection detection algorithms) where accurate system solution determines feasible states in multidimensional constraint spaces. The distractors strategically target pervasive error patterns including single-equation satisfaction without dual verification, arithmetic miscalculation during division steps, and confusion between individual equation features versus system solutions highlighting the necessity of explicit verification protocols requiring substitution into every original equation before accepting any candidate solution, a disciplined habit preventing undetected errors that compromise solution validity in increasingly complex systems with three or more variables where verification becomes exponentially more critical.