This question tests graphing proportional relationships y=kx, interpreting slope as unit rate, and comparing relationships from different representations. Proportional relationships have the form y=kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Comparing: steeper slope or larger k indicates greater rate (y=5x faster than y=2x since 5>2). For example, in y=9x, for x=4, y=9*4=36 dollars. The correct cost is $36 because it applies the unit rate of $9 per hour correctly for 4 hours. A common error is misapplying the equation, like adding instead of multiplying or using wrong k. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Mistakes: forgetting origin requirement, inverting slope, comparing wrong values (using y-intercept when proportional has none).

This question tests graphing proportional relationships of the form y = kx, interpreting the slope as the unit rate, and comparing relationships from different representations. Proportional relationships have the form y = kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Comparing: steeper slope or larger k indicates greater rate (y=5x faster than y=2x since 5>2). Here, for every 1 cup of water (x), 7 drops of dye (y) are added, so y=7x and slope 7 means 7 drops per cup. The correct choice B properly interprets the unit rate in context as drops per cup. A common error is reversing the units, like claiming cups per drop in choice A. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Mistakes: forgetting origin requirement, inverting slope, comparing wrong values (using y-intercept when proportional has none).

This question tests graphing proportional relationships y=kx, interpreting slope as unit rate, and comparing relationships from different representations. Proportional relationships have the form y=kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Comparing: steeper slope or larger k indicates greater rate (y=5x faster than y=2x since 5>2). For example, the line passes through (3,12), so k=12/3=4. The correct answer is k=4 because it is properly calculated as the slope rise/run from (0,0) to (3,12). A common error is inverting to k=3/12 or using x instead of y/x. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Mistakes: forgetting origin requirement, inverting slope, comparing wrong values (using y-intercept when proportional has none).

This question tests finding the unit rate from a proportional relationship given one data point. Proportional relationships have form y=kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Given that 3 packs contain 24 picks, we find the unit rate by dividing: 24 picks ÷ 3 packs = 8 picks per pack. The correct answer is A because the constant ratio y/x = 24/3 = 8 picks per pack. Common errors include B (21), C (27), or D (72) which might result from arithmetic mistakes or using the wrong operation (like 24-3=21 or 24×3=72). Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Unit rate always means "amount of y per one unit of x."

This question tests identifying which equation represents a proportional relationship from its form. Proportional relationships have form y=kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Among the choices, only y=7x has the form y=kx with k=7, making it proportional. The correct answer is B because y=7x passes through the origin (when x=0, y=0) and has constant ratio y/x=7. Common errors include choosing A (y=2x+3) or D (y=5+x) which have non-zero y-intercepts, or C (y=x²) which is nonlinear. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Any added constant or exponent other than 1 makes the relationship non-proportional.

This question tests graphing proportional y=kx, interpreting slope as unit rate, and comparing relationships from different representations. Proportional relationships have form y=kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Comparing: steeper slope or larger k indicates greater rate (y=5x faster than y=2x since 5>2). In the equation y=6x, the slope 6 means the distance increases by 6 miles for every 1 hour. The correct choice is B because it accurately interprets the slope as the unit rate of 6 miles per hour. A common error is inverting the units, like in A, saying 6 hours per mile. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Mistakes: forgetting origin requirement, inverting slope, comparing wrong values (using y-intercept when proportional has none).

This question tests graphing proportional y=kx, interpreting slope as unit rate, and comparing relationships from different representations. Proportional relationships have form y=kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Comparing: steeper slope or larger k indicates greater rate (y=5x faster than y=2x since 5>2). The line through (0,0) and (2,10) has slope rise/run=10/2=5. The correct choice is B because it calculates the slope correctly as 10/2=5. A common error is inverting the slope, like in A, using 2/10=0.2. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Mistakes: forgetting origin requirement, inverting slope, comparing wrong values (using y-intercept when proportional has none).

This question tests graphing proportional relationships y=kx, interpreting slope as unit rate, and comparing relationships from different representations. Proportional relationships have the form y=kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Comparing: steeper slope or larger k indicates greater rate (y=5x faster than y=2x since 5>2). For example, in y=7x, the slope k=7 means y increases by 7 for every 1 unit increase in x. The correct choice explains that for every 1 unit increase in x, y increases by 7 units, properly interpreting the slope as the unit rate. A common error is inverting the slope to say for every 7 units in x, y increases by 1, which reverses the rise over run. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Mistakes: forgetting origin requirement, inverting slope, comparing wrong values (using y-intercept when proportional has none).

This question tests graphing proportional relationships y=kx, interpreting slope as unit rate, and comparing relationships from different representations. Proportional relationships have the form y=kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Comparing: steeper slope or larger k indicates greater rate (y=5x faster than y=2x since 5>2). For example, in y=2x+3, when x=0, y=3, so it does not pass through (0,0). The correct explanation is that it is incorrect because proportional relationships must pass through (0,0), and this one has a y-intercept of 3. A common error is claiming it's proportional just because y increases with x or focusing on slope alone. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Mistakes: forgetting origin requirement, inverting slope, comparing wrong values (using y-intercept when proportional has none).

This question tests graphing proportional relationships of the form y = kx, interpreting the slope as the unit rate, and comparing relationships from different representations. Proportional relationships have the form y = kx (passes through origin, k is constant rate): graphed as straight line through (0,0) with slope k (rise/run ratio constant), interpreted as unit rate (k miles per hour, k dollars per item, k cups per serving). Comparing: steeper slope or larger k indicates greater rate (y=5x faster than y=2x since 5>2). The line passes through (0,0) and (4,20), so slope k = 20/4 = 5, giving y=5x. The correct choice B properly calculates the slope from the points and forms the equation. A common error is inverting the slope to 1/5 or adding an intercept. Strategy: (1) check origin (proportional must pass through (0,0)), (2) find slope/rate (from graph: rise/run, from table: y/x for any point, from equation: coefficient of x), (3) compare if multiple (larger k or steeper slope wins), (4) interpret (slope 60 in distance-time means 60 miles per hour). Mistakes: forgetting origin requirement, inverting slope, comparing wrong values (using y-intercept when proportional has none).