This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Here, the verbal description states 3 cups of flour per batch, so k=3 cups per batch in y=3x. Common errors include inverting to 1/3, using multiples like 6 or 9, or misinterpreting "per batch" as x instead of the rate. From verbal descriptions, the stated rate is k ("3 meters per second" → k=3 m/s); to find k from a table, pick any (x,y) pair, calculate k=y/x, verify with others. Special point (1,k): proportional graphs pass through (1,k) where k is constant—makes k directly readable; not proportional if y-intercept ≠0, no k value.

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Here, the graph passes through the origin and (1,3), so k=3 directly from the y-coordinate at x=1, or slope=3/1=3. Common errors include using the inverse like 1/3, confusing with other points, or thinking the slope is 4 if misreading. To find k from a graph, use slope=rise/run through the origin, or read y at x=1 giving (1,k) where k is that y-value; from an equation y=kx, k is the coefficient. Proportional graphs pass through (1,k) where k is the constant—making k directly readable (no calculation needed, just read the y-coordinate at x=1); mistakes include calculating ratios wrong (x/y not y/x) or reading the graph at the wrong point.

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Relationship 1 has k=6 from y=6x; Relationship 2 has k=4 from table ratios (4/1=4, 8/2=4, 12/3=4), so Relationship 1 has greater k since 6>4. Common errors include miscalculating table ratios (e.g., x/y=1/4), thinking they are equal, or inverting. To find k, from equation it's the coefficient, from table calculate y/x and verify; special point (1,k) for graphs. Mistakes: assuming not enough info when data is given, or confusing with non-proportional cases.

This skill tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship $y=kx$ has constant k equal to: (1) ratio y/x for any point ($14/2=7$, $28/4=7$, $k=7$), (2) slope of graph ($rac{\text{rise}}{\text{run}}$ through origin), (3) coefficient of x in equation ($y=7x \to k=7$), (4) unit rate stated ("7 dollars per item" $\to k=7$). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). In this verbal description, the car travels 65 miles per gallon, so with y miles and x gallons, k=65 as the stated unit rate in $y=65x$. Common mistakes include inverting to 1/65 (A), using decimals like 0.65 (C) or multiplying unnecessarily to 650 (D). Finding k: from table (pick any (x,y) pair, calculate k=y/x, verify with other pairs—should all equal), from graph (slope=\frac{\text{rise}}{\text{run}}$ through origin, or read y at x=1 giving (1,k), k is that y-value), from equation y=kx (k is coefficient: $y=7x \to k=7$), from verbal (stated rate is k: "3 meters per second" $\to k=3$ m/s). Not proportional: if y-intercept≠0 (line misses origin), no constant of proportionality exists ($y=mx+b$ with b≠0 is linear but not proportional, no k value).

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, table showing x:2,4,6 y:14,28,42, calculate ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or graph through (0,0) and (1,7) has k=7 from point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. In the equation y=4.5x, k is the coefficient 4.5. Errors include mistaking it for 1/4.5 (inverting) or confusing with non-proportional forms like y=4.5x+something. From an equation y=kx, k is simply the coefficient of x; if there's a y-intercept (b≠0), it's not proportional and has no k.

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. In the equation y=(3/2)x, k=3/2 is the coefficient. Common errors include simplifying incorrectly to 2/3, using numerator 3 or denominator 2 alone, or inverting. From an equation y=kx, k is the coefficient; special point (1,k): graphs pass through (1,k) where k=3/2 here. Mistakes: confusing with non-proportional (y=mx+b, b≠0 has no k), or calculating ratios wrong.

This skill tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). Since the line passes through (1,9) and is proportional (through origin), k=9 directly as the y-value at x=1, or slope=9/1=9. Common errors include choosing 1 (A) as x-value, inverting to 1/9 (C), or mistaking for 10 (D). Finding k: from table (pick any (x,y) pair, calculate k=y/x, verify with other pairs—should all equal), from graph (slope=rise/run through origin, or read y at x=1 giving (1,k), k is that y-value), from equation y=kx (k is coefficient: y=7x → k=7), from verbal (stated rate is k: "3 meters per second" → k=3 m/s). Mistakes: confusing slope with y-intercept (using b as k), calculating ratios wrong (x/y not y/x), reading graph at wrong point, non-proportional relationships claimed to have k.

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, table showing x:2,4,6 y:14,28,42, calculate ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or graph through (0,0) and (1,7) has k=7 from point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. The description is 65 miles per gallon, so k=65 (miles per gallon). Mistakes: inverting to 1/65, or misapplying like 65/2 for no reason. From verbal, k is the rate: "65 miles per gallon" → k=65; confirm proportionality (y directly varies with x without constants).

This skill tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). In this graph passing through (0,0) and (2,10), the slope is rise/run=10/2=5, so k=5, or equivalently y/x=10/2=5. Errors include using the y-value 10 directly (A), inverting to 2/10=1/5 (B), or picking unrelated numbers like 8 (D). Finding k: from table (pick any (x,y) pair, calculate k=y/x, verify with other pairs—should all equal), from graph (slope=rise/run through origin, or read y at x=1 giving (1,k), k is that y-value), from equation y=kx (k is coefficient: y=7x → k=7), from verbal (stated rate is k: "3 meters per second" → k=3 m/s). Special point (1,k): proportional graphs pass through (1,k) where k is constant—makes k directly readable (no calculation needed, just read y-coordinate at x=1).

This skill tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). In this question, the line passes through the origin and (1,3), so k is directly the y-value at x=1, giving k=3 as the slope or unit rate. Common errors include mistaking k for the x-value (choosing 1 for A), inverting to x/y=1/3 (B), or using another point incorrectly like assuming k=4 without basis (D). Finding k: from table (pick any (x,y) pair, calculate k=y/x, verify with other pairs—should all equal), from graph (slope=rise/run through origin, or read y at x=1 giving (1,k), k is that y-value), from equation y=kx (k is coefficient: y=7x → k=7), from verbal (stated rate is k: "3 meters per second" → k=3 m/s). Special point (1,k): proportional graphs pass through (1,k) where k is constant—makes k directly readable (no calculation needed, just read y-coordinate at x=1).