This question tests verifying collinearity by constant slope, where any pairs give same m, using similar triangles to explain, and relating to y=mx+b derivation. Slope m=(y₂-y₁)/(x₂-x₁) constant on lines; similar triangles prove via equal angles (parallel), proportional sides like 6/2=12/4=3; derive y=mx+b with m from points, b from intercept. Slopes: (12-4)/(5-1)=8/4=2, (8-4)/(3-1)=4/2=2. Matches A showing same slope. Errors: inverting to 1/2 in B, misstating second as 4 in C, inverting formula in D. Process: (1) calculate m for pairs, (2) verify equality, (3) find b if needed via point or intercept, (4) equation optional here. Similar triangles: shared angle, proportional sides, constant ratio; mistakes: inverting rise/run, claiming variation, wrong form.

This question tests deriving a line equation from points, ensuring constant slope where any two give same m, using similar triangles for constancy, and forming y=mx+b. Slope m=(y₂-y₁)/(x₂-x₁) constant; similar triangles with equal angles (parallel sides), proportional like 6/2=12/4=3; y=mx+b with m from points, b from (0,b). Using (0,4),(2,8), m=(8-4)/(2-0)=4/2=2, b=4; check (4,12): 2*4+4=12. Equation y=2x+4 matches A. Errors: swapping to 4x+2 in B, -4 in C, 1/2 in D. Steps: (1) compute m from 2 points, (2) verify with third, (3) non-origin find b from intercept, (4) equation. Similar triangles: shared angle, proportionality, constant rise/run; mistakes: inverting, variation claim, wrong form.

This question tests your understanding of deriving the equation of a line using its constant slope, where any two points give the same m, and incorporating similar triangles to explain constancy while forming y=mx for origin lines or y=mx+b for others. The slope m = (y₂ - y₁)/(x₂ - x₁) is constant for any point pairs on the line, a key feature of straight lines; similar triangles demonstrate this as triangles on the same line share angles from parallel sides and have proportional sides, like 2× rise and 2× run yielding 6/2 = 12/4 = 3, confirming constant slope; for y=mx through the origin, m = y/x from any point, or for y=mx+b, find m from two points and b from the y-intercept (0,b). Here, using points (0,2) and (3,8), m = (8-2)/(3-0) = 6/3 = 2, and since it crosses y at 2, b=2. This gives the equation y=2x+2, matching choice B. Errors include wrong slope like 3 in A or 1/2 in D, or incorrect b like -2 in C. The process is: (1) pick 2 points to calculate m = (y₂ - y₁)/(x₂ - x₁), (2) verify if needed with another pair, (3) for non-origin lines, find b by noting y-intercept or solving y=mx+b with a point, (4) write the equation. Similar triangles show proof via shared angles and proportional sides ensuring constant rise/run; common mistakes are inverting rise/run, assuming varying slope, or using y=mx for non-origin lines.

This question tests deriving a line equation from points, ensuring constant slope where any two give same m, using similar triangles for constancy, and forming y=mx+b. Slope m=(y₂-y₁)/(x₂-x₁) constant; similar triangles with equal angles (parallel sides), proportional like 6/2=12/4=3; y=mx+b with m from points, b from (0,b). Using (0,4),(2,8), m=(8-4)/(2-0)=4/2=2, b=4; check (4,12): 2*4+4=12. Equation y=2x+4 matches A. Errors: swapping to 4x+2 in B, -4 in C, 1/2 in D. Steps: (1) compute m from 2 points, (2) verify with third, (3) non-origin find b from intercept, (4) equation. Similar triangles: shared angle, proportionality, constant rise/run; mistakes: inverting, variation claim, wrong form.

This question tests deriving a line equation with given y-intercept and a point, using constant slope concept where any points give same m, similar triangles for constancy, and forming y=mx+b. Slope m=(y₂-y₁)/(x₂-x₁) constant on lines; similar triangles have equal angles (parallel sides) and proportional sides, e.g., 6/2=12/4=3, proving constant; derive y=mx+b with b given, m from point: plug into y=mx+b solve for m. With b=5 and (2,9), 9 = m*2 + 5, so m= (9-5)/2 = 4/2 = 2. Equation y=2x+5 matches choice A. Errors: wrong m like 1/2 in B, -5 b in C, or 4 in D. Steps: (1) if needed compute m from points, (2) verify constancy, (3) with b known, solve for m using point in y=mx+b, (4) write equation. Similar triangles: shared angle, proportionality from parallelism, constant ratio; mistakes: inverting, assuming variation, wrong form.

This question tests comparing slopes for steepness, understanding constant m on each line from any points, using similar triangles to explain, and deriving y=mx or y=mx+b. Slope m=(y₂-y₁)/(x₂-x₁) constant per line; similar triangles show equal angles (parallel) and proportional sides, like 6/2=12/4=3; y=mx for origin, y=mx+b otherwise with m from points, b from intercept. Line 1: (0,0),(3,6), m=6/3=2; Line 2: (0,2),(3,5), m=3/3=1. Line 1 steeper with larger |m|=2 >1, as in B. Errors: swapping in A, claiming equal via wrong calc in C, or y-intercept in D. Process: (1) calculate m per line, (2) verify if needed, (3) origin y=mx else find b, (4) equation or compare. Similar triangles proof: shared angle, proportional sides, constant ratio; mistakes: inverting, varying claim, wrong form.

This question tests verifying constant slope on a line, where any two points give the same m, using similar triangles for explanation, and linking to deriving y=mx or y=mx+b equations. Slope m = (y₂ - y₁)/(x₂ - x₁) remains constant, defining lines; similar triangles prove this with equal angles from parallel sides and proportional sides, e.g., 6/2 = 12/4 = 3, showing constant slope; derive y=mx for origin via m=y/x, or y=mx+b with m from points and b from (0,b). For points (2,5),(4,9),(6,13), m between first two: (9-5)/(4-2)=4/2=2, between last two: (13-9)/(6-4)=4/2=2. This confirms constant slope as in choice A. Common errors: inverting to 1/2 in B, stating 4 without dividing in C, or miscalculating second as 4 in D. Steps: (1) pick 2 points, calculate m, (2) verify with another pair, (3) if origin y=mx, else find b via intercept or solving with point, (4) equation. Similar triangles: shared angle, proportional sides from parallelism, constant ratio; mistakes: inverting rise/run, claiming variation, wrong form for origin/non-origin.

This question tests verifying collinearity by constant slope, where any pairs give same m, using similar triangles to explain, and relating to y=mx+b derivation. Slope m=(y₂-y₁)/(x₂-x₁) constant on lines; similar triangles prove via equal angles (parallel), proportional sides like 6/2=12/4=3; derive y=mx+b with m from points, b from intercept. Slopes: (12-4)/(5-1)=8/4=2, (8-4)/(3-1)=4/2=2. Matches A showing same slope. Errors: inverting to 1/2 in B, misstating second as 4 in C, inverting formula in D. Process: (1) calculate m for pairs, (2) verify equality, (3) find b if needed via point or intercept, (4) equation optional here. Similar triangles: shared angle, proportional sides, constant ratio; mistakes: inverting rise/run, claiming variation, wrong form.

This question tests understanding of constant slope on a line through the origin, where any two points yield the same m, using similar triangles for explanation, and deriving y=mx specifically for origin-passing lines. Slope m = (y₂ - y₁)/(x₂ - x₁) stays constant across point pairs, defining straight lines; similar triangles illustrate this with equal angles from parallel sides and proportional dimensions, such as a doubled triangle giving 6/2 = 12/4 = 3, proving constancy; for origin lines, y=mx with m = y/x from any point, or generally y=mx+b with b=0. For (0,0) and (4,12), m = (12-0)/(4-0) = 12/4 = 3. Thus, the equation is y=3x, as in choice C. Mistakes include using y=12x in A, adding b=12 in B despite origin, or inverting to 1/3 in D. Steps: (1) choose 2 points, compute m = (y₂ - y₁)/(x₂ - x₁), (2) check another pair if available, (3) since through origin, use y=mx, (4) write equation. Similar triangles confirm via shared angles and proportionality, keeping rise/run constant; errors like inverting rise/run, claiming variable slope, or applying y=mx+b unnecessarily.

This question tests calculating constant slope between two points on a line, understanding any pair gives same m, with similar triangles explaining constancy, and deriving y=mx or y=mx+b. Slope m = (y₂ - y₁)/(x₂ - x₁) is constant for lines; similar triangles show equal angles (parallel sides) and proportional sides, like 6/2=12/4=3, proving constancy; y=mx for origin via m=y/x, y=mx+b using m from points and b from intercept. For (2,1) and (6,9), m = (9-1)/(6-2) = 8/4 = 2. This matches choice B with correct order and positive 2. Errors: wrong sign or order giving -2 in A/D, inverting to 1/2 in C. Process: (1) select points, compute m=(y₂-y₁)/(x₂-x₁), (2) verify if more points, (3) origin y=mx, else b from intercept or point plug-in, (4) equation. Similar triangles proof: shared angle, proportional sides, constant rise/run; mistakes: inverting, varying slope, incorrect form.