Solve for x: 5(x - 3) = 2x + 9

The variable x equals 8 when the distributive property is first applied to expand 5(x - 3) into 5x - 15, the equation is rearranged to consolidate variable terms on one side and constants on the other yielding 3x = 24, and finally division by 3 isolates x = 8 as the solution satisfying the original equation upon verification.

A. 2

Substituting x = 2 produces 5(2 - 3) = 5(-1) = -5 on the left versus 2(2) + 9 = 4 + 9 = 13 on the right revealing an 18-unit disparity with opposite signs that completely violates equation equality. This error likely originates from solving 5x - 15 = 2x + 9 by subtracting 2x to get 3x - 15 = 9, then incorrectly subtracting 15 from 9 to obtain 3x = -6 → x = -2, followed by sign omission during transcription to yield x = 2. Another plausible pathway involves distributing incorrectly as 5x - 3 = 2x + 9 (omitting multiplication of 5 × 3), then solving 3x = 12 → x = 4 but mis-transcribing as 2. The dramatic failure upon verification (-5 versus 13) reveals fundamental procedural errors in either distribution execution or sign management during constant transposition deficiencies requiring explicit reinforcement of distributive property mechanics (multiplying coefficient across all terms within parentheses) and the principle that transposing terms across equality requires sign inversion to maintain balanced equivalence.

B. 4

This value yields 5(4 - 3) = 5(1) = 5 versus 2(4) + 9 = 8 + 9 = 17 upon substitution producing values with 12-unit separation despite both being positive. The error likely originates from incomplete distribution: executing 5(x) - 3 = 5x - 3 instead of 5x - 15, then solving 5x - 3 = 2x + 9 → 3x = 12 → x = 4. This "partial distribution" error represents a critical misconception where learners apply the distributive coefficient to the variable term but omit multiplication with the constant term within parentheses a pervasive deficiency particularly common when negative signs appear inside parentheses creating cognitive load that triggers selective application of the distributive operation. Students might recognize 5 must multiply x but mentally "drop" the -3 during distribution due to sign complexity, executing 5·x = 5x correctly but treating -3 as already outside the parentheses. This distractor effectively identifies learners with fragmented distributive property understanding who lack systematic "multiply coefficient by every term inside parentheses" protocols a deficiency requiring explicit visual scaffolding using color-coding or underlining to ensure all parenthetical terms receive coefficient multiplication before proceeding to equation solving steps.

C. 6

Substituting x = 6 yields 5(6 - 3) = 5(3) = 15 versus 2(6) + 9 = 12 + 9 = 21 revealing a consistent 6-unit deficit that suggests systematic miscalculation during constant combination. The error likely originates from correctly distributing to obtain 5x - 15 = 2x + 9 and subtracting 2x to get 3x - 15 = 9, but then adding 9 instead of 15 to both sides (3x = 18 → x = 6) through inverse operation confusion treating the -15 as requiring addition of 9 rather than 15 for elimination. Students might execute "move the 9 over" by adding 9 to both sides yielding 3x - 6 = 18, then incorrectly solving 3x = 24 → x = 8 but mis-transcribing as 6 through digit substitution. Another plausible pathway involves solving 5x - 2x = 9 + 15 → 3x = 24 correctly but then dividing 24 ÷ 4 = 6 instead of 24 ÷ 3 = 8 through divisor misreading. The consistent 6-unit error magnitude (21 - 15 = 6) corresponds precisely to the difference between correct and incorrect constant combination (15 versus 9), revealing procedural rather than conceptual deficiency requiring emphasis on systematic term transposition with explicit sign tracking during multi-step algebraic manipulation.

D. 8

This value correctly satisfies the equation through systematic algebraic manipulation executed via multiple verification pathways. Distribution step: 5(x - 3) = 5x - 15. Equation setup: 5x - 15 = 2x + 9. Variable consolidation: subtract 2x from both sides → 3x - 15 = 9. Constant elimination: add 15 to both sides → 3x = 24. Isolation: divide by 3 → x = 8. Verification substitution: left side 5(8 - 3) = 5(5) = 25; right side 2(8) + 9 = 16 + 9 = 25; equality confirmed. Alternative solution pathway: 5x - 2x = 9 + 15 → 3x = 24 → x = 8, demonstrating efficient term grouping without intermediate steps. Graphical interpretation: the lines y = 5(x - 3) and y = 2x + 9 intersect at x = 8, confirming solution as intersection point. This solution demonstrates comprehensive mastery of linear equation solving including distributive property application with sign preservation, term consolidation strategies (grouping variables on one side, constants on the other), inverse operation sequencing with sign management, and solution verification through substitution integrated competencies essential for algebraic modeling, systems of equations, function analysis, and real-world constraint satisfaction where accurate solution determination informs decision-making across scientific, economic, and engineering contexts requiring precise quantitative resolution.

Conclusion

The solution x = 8 emerges through rigorous application of distributive property followed by systematic term consolidation and isolation maintaining equality balance throughout transformation sequences. This problem reinforces critical algebraic manipulation competencies essential across mathematical domains: executing complete distribution across all parenthetical terms without omission, recognizing efficient term grouping strategies that minimize intermediate steps (moving all variables to one side, constants to the other simultaneously), managing sign changes during transposition with precision, and verifying solutions through original equation substitution to detect procedural errors introduced during multi-step manipulation. Mastery of these integrated skills proves indispensable for calculus (limit evaluation requiring algebraic simplification), physics (equation rearrangement for unknown isolation), economics (supply-demand equilibrium solving), and computer science (algorithmic constraint resolution) where accurate equation solving forms the backbone of quantitative problem resolution. The distractors strategically target pervasive error patterns including partial distribution, inverse operation confusion, and arithmetic miscalculation during constant combination highlighting the necessity of explicit distributive property protocols ("multiply coefficient by every term inside parentheses") and verification habits before final answer submission to prevent undetected errors that compromise solution validity in increasingly complex algebraic contexts.