Which of the following is the equation of a line with slope 3 passing through (2, 5)?

The equation y = 3x - 1 correctly represents the line with slope 3 passing through (2, 5) when the point-slope formula y - y₁ = m(x - x₁) is applied using m = 3 and (x₁, y₁) = (2, 5), yielding y - 5 = 3(x - 2) which simplifies to y = 3x - 6 + 5 = 3x - 1, verified by substituting x = 2 to obtain y = 5.

A. y = 3x - 1

This equation correctly satisfies both slope and point conditions through systematic derivation and comprehensive verification. Point-slope derivation: y - 5 = 3(x - 2) → y - 5 = 3x - 6 → y = 3x - 1. Point verification: substitute x = 2 → y = 3(2) - 1 = 6 - 1 = 5 ✓. Slope verification: coefficient of x is 3 ✓. Alternative point testing: x = 0 → y = -1 (y-intercept); x = 1 → y = 2; x = 3 → y = 8 each successive x-increase of 1 yields y-increase of 3, confirming constant slope 3. Graphical interpretation: line crosses y-axis at (0, -1) and passes through (2, 5), with rise 3 over run 1 between consecutive integer points. This solution demonstrates comprehensive mastery of linear equation formulation including point-slope formula application, algebraic simplification maintaining equality, slope-intercept conversion, and dual verification through point substitution and slope inspection integrated competencies essential for coordinate geometry, calculus (tangent line determination), physics (position-time relationships), and data modeling (trend line fitting) where accurate line equation derivation enables prediction, interpolation, and relationship quantification across scientific and analytical contexts requiring precise linear modeling.

B. y = 3x + 1

This equation has correct slope 3 but fails point verification: substituting x = 2 yields y = 3(2) + 1 = 7 ≠ 5 producing a y-value 2 units too high. The error likely originates from sign mishandling during point-slope simplification: executing y - 5 = 3(x - 2) → y - 5 = 3x - 6 correctly but then adding 5 to both sides as y = 3x - 6 - 5 = 3x - 11 (option C) or incorrectly as y = 3x + 6 + 5 = 3x + 11 (option D), with this option representing an intermediate miscalculation where -6 + 5 was computed as +1 instead of -1. Students might execute -6 + 5 = -1 correctly but then mis-transcribe the negative sign during answer selection. Another plausible pathway involves using incorrect point (2, 7) through y-coordinate misreading. The consistent 2-unit vertical offset (7 versus 5) corresponds precisely to the sign error magnitude in constant term calculation (-1 versus +1 differ by 2), revealing arithmetic rather than conceptual deficiency requiring emphasis on careful sign management during algebraic simplification and mandatory point verification before final answer selection to catch transcription errors that preserve slope correctness while distorting positional accuracy particularly dangerous as slope-only verification would miss this error, producing lines with correct steepness but incorrect placement.

C. y = 3x - 11

This equation has correct slope 3 but fails point verification: substituting x = 2 yields y = 3(2) - 11 = 6 - 11 = -5 ≠ 5 producing a y-value 10 units too low with opposite sign. The error likely originates from sign inversion during constant combination: executing y - 5 = 3(x - 2) → y - 5 = 3x - 6 correctly but then adding 5 to the right side as -6 - 5 = -11 instead of -6 + 5 = -1 treating the transposition as subtraction rather than addition. Students might misapply the rule "move term to other side and change sign" by changing the sign of the 5 being moved but forgetting that moving -5 from left to right requires adding 5 to both sides, not subtracting. Another plausible pathway involves solving for b in y = mx + b using 5 = 3(2) + b → 5 = 6 + b → b = -1 but then misreading b = -1 as b = -11 through digit duplication. The dramatic 10-unit error magnitude (5 versus -5) reveals systematic sign management deficiency during equation rearrangement a critical procedural gap requiring explicit emphasis on balanced operation execution ("add 5 to both sides" rather than "move and change sign") to prevent sign inversion errors that fundamentally distort line positioning while preserving slope characteristics.

D. y = 3x + 11

This equation has correct slope 3 but fails point verification: substituting x = 2 yields y = 3(2) + 11 = 6 + 11 = 17 ≠ 5 producing a y-value 12 units too high. The error likely originates from double sign error during simplification: executing y - 5 = 3(x - 2) → y - 5 = 3x - 6 correctly but then adding 5 to the right side as -6 + 5 = -1, then erroneously changing the sign to +11 through unstructured adjustment. Another plausible pathway involves solving 5 = 3(2) + b → 5 = 6 + b → b = -1 but then executing -1 × -11 = 11 through multiplicative confusion. Students might recognize the constant should be negative but misremember magnitude as 11 instead of 1, possibly influenced by the x-coordinate value 2 appearing in calculation steps. The 12-unit excess (17 - 5 = 12) equals precisely twice the x-coordinate value (2 × 6 = 12) suggesting possible coefficient misapplication where students executed 3 × 2 × 2 = 12 then added to 5 to reach 17. This distractor effectively identifies learners with fragmented equation derivation knowledge who recognize slope-intercept form requirements but lack systematic constant determination protocols, producing solutions with correct slope but wildly inaccurate intercepts a dangerous error pattern requiring explicit substitution verification before final answer submission to catch magnitude errors that compromise predictive accuracy in modeling contexts where line position determines outcome values.

Conclusion

The equation y = 3x - 1 emerges through rigorous application of point-slope formula followed by careful algebraic simplification with sign preservation, verified through point substitution confirming both coordinates satisfy the equation. This problem reinforces critical linear equation competencies essential across mathematical domains: selecting appropriate formulation method based on given information (point-slope when point and slope known), executing algebraic manipulation with precise sign management during transposition, converting between equation forms while preserving geometric meaning, and verifying solutions through substitution to detect procedural errors introduced during simplification. Mastery of these integrated skills proves indispensable for calculus (derivative as slope enabling tangent line equations), physics (kinematic equations modeling motion), statistics (regression line derivation), and computer graphics (line rendering algorithms) where accurate line equation determination enables prediction, interpolation, and spatial relationship modeling across scientific and technical applications requiring precise linear representation. The distractors strategically target pervasive error patterns including sign mishandling during constant combination, arithmetic errors in integer operations, and transcription mistakes preserving slope while distorting intercept highlighting the necessity of mandatory point verification protocols after equation derivation to catch errors that slope-only inspection would miss, ensuring both directional and positional accuracy in linear models with real-world predictive implications.