Both students correctly identify cost $$C$$ as the dependent variable. Student A's equation $$C = 30d + 0.25m$$ accounts for multiple independent variables (days $$d$$ and miles $$m$$). Student B's equation $$C = 30 + 0.25m$$ is valid for single-day analysis with miles as the independent variable. Choice C recognizes that equations can have multiple independent variables.

Temperature is independent (researchers control it), volume is dependent (responds to temperature). Rate is 15 mL per 10°C = 1.5 mL per °C. Since volume is 200 mL at 20°C, the equation is $$V = 200 + 1.5(T - 20)$$. Choice A ignores the reference temperature. Choices C and D incorrectly make volume independent, which contradicts experimental control.

When working with equations that model real-world relationships, you need to identify what information you're given and what you're trying to find. This determines which variable you'll substitute and which one you'll solve for.

In this plant growth problem, the equation $$h = 15 + 2.5d$$ tells you that height depends on the number of days. You're asked when the plant will reach 35 cm, meaning you know the target height (35 cm) and want to find the corresponding time in days.

The correct approach is D: substitute $$h = 35$$ and solve for $$d$$. Since you know the height and need to find the days, you'd write $$35 = 15 + 2.5d$$, then solve: $$20 = 2.5d$$, so $$d = 8$$ days.

A is incorrect because you don't rewrite the original equation's structure. The relationship $$h = 15 + 2.5d$$ correctly shows height depending on days, which matches your problem setup.

B reverses the roles of the variables. You're not given days (35) to find height—you're given a target height to find the days needed.

C treats both variables as if they're given values, but the question asks "when will the plant reach 35 cm?" This means 35 is your target height, not an input for calculating something else.

Study tip: In word problems involving equations, always identify "what do I know?" versus "what am I looking for?" The known value gets substituted, and you solve for the unknown variable.

The number of hours $$h$$ is the independent variable because it can be chosen freely, while the cost $$C$$ depends on the movie length. The equation is $$C = 3h + 5$$ because there's a $5 base fee plus $3 per hour. Choice A reverses the coefficients. Choices C and D incorrectly identify which variable is independent.

The student correctly identified the variables and equation. Time $$m$$ is independent (you choose when to measure), and gallons $$g$$ is dependent (determined by how much time has passed). The equation $$g = 120 - 8m$$ correctly shows 120 initial gallons minus 8 gallons per minute. Choices B, C, and D suggest incorrect modifications to a properly written equation.

This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (time t we jog—we control), dependent is output/depends on independent (distance d depends on time—results from independent). Equation: express dependent in terms of independent (d=6t: distance equals 6 times time, dependent d on left, independent t in expression). Table: list independent values (t: 0,1,2,3), calculate dependent using equation (if t=1, d=6×1=6; t=2, d=12; etc.). Graph: independent on x-axis (horizontal: time), dependent on y-axis (vertical: distance), plot ordered pairs ((1,6), (2,12),...), proportional d=kt graphs through origin. For example, distance-time at 65 mph, independent=time t (hours driven), dependent=distance d (miles traveled), equation d=65t, table t:1,2,3 d:65,130,195 (each from 65×t), graph: x-axis time, y-axis distance, points (1,65),(2,130),(3,195) forming line through (0,0) with slope 65 matching equation coefficient. The correct equation is d=6t, showing distance as a function of time at constant speed. Errors like d=t+6 (additive instead of multiplicative) or d=t/6 (inverse) don't match the proportional relationship. Analyzing: (1) identify relationship (distance depends on time at 6 mph), (2) determine independent (time t) and dependent (distance d), (3) write equation (d=6t), (4) create table (t:0,1,2, d:0,6,12), (5) graph (x=t, y=d, plot pairs, line through origin), (6) connect (equation matches table: 6×1=6✓, slope=6✓). Axes convention: independent horizontal (x-axis: time), dependent vertical (y-axis: distance).

This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (hours h worked—we control), dependent is output/depends on independent (money m depends on hours—results from independent). Equation: express dependent in terms of independent (m=5h: money equals 5 times hours, dependent m on left, independent h in expression). Table: list independent values (h: 0,1,2,3), calculate dependent using equation (if h=0, m=0; h=1, m=5; etc.). Graph: independent on x-axis (horizontal: hours), dependent on y-axis (vertical: money), plot ordered pairs ((0,0), (1,5),...), proportional through origin. For example, earning $4 per chore, independent=chores c, dependent=earnings e, equation e=4c, table c:0,1,2 e:0,4,8, graph: x-chores, y-earnings, points (0,0),(1,4),(2,8) line through origin slope 4. The correct ordered pairs are (0,0),(1,5),(2,10),(3,15), with (h,m) format. Errors like reversing pairs ((5,1) instead of (1,5)) or starting without (0,0) for proportional. Analyzing: (1) identify relationship (money depends on hours at $5 each), (2) determine independent (h) and dependent (m), (3) write equation (m=5h), (4) create table (h:0,1,2,3; m:0,5,10,15), (5) graph (x=h, y=m, plot pairs), (6) connect (pairs match equation: 5*1=5✓). Axes convention: independent horizontal (x-axis: hours), dependent vertical (y-axis: money).

Tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (hours h we rent, items n we buy—we control), dependent is output/depends on independent (total cost c depends on hours, cost depends on items—results from independent). Equation: express dependent in terms of independent (c=3+2h: cost equals 3 plus 2 times hours, dependent c on left, independent h in expression). Table: list independent values (h: 0,1,2,3), calculate dependent using equation (if h=1, c=3+2×1=5; h=2, c=7; etc.). Graph: independent on x-axis (horizontal: hours), dependent on y-axis (vertical: cost), plot ordered pairs ((1,5), (2,7),...), non-proportional due to intercept. In this bike rental example, independent is hours rented (you choose), dependent is total cost (results from choice plus fees), equation c=2h+3, table h:0,1,2 c:3,5,7, graph x=hours, y=cost, line with slope 2 and y-intercept 3. The correct identification is independent: hours rented; dependent: total cost, as in choice B, with others reversing or misusing constants.

This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (time t we travel), dependent is output/depends on independent (distance d depends on time). Equation: express dependent in terms of independent (d=50t: distance equals 50 times time, dependent d on left, independent t in expression). Table: list independent values (t: 0,1,2,3), calculate dependent using equation (if t=0, d=0; t=1, d=50; t=2, d=100; t=3, d=150). Graph: independent on x-axis (horizontal: time), dependent on y-axis (vertical: distance), plot ordered pairs ((0,0), (1,50),...), proportional through origin. Choice B correctly shows the table with d=50t values starting from 0. Errors like A (starting d=50 at t=0, not 0), C (t starts at 1, d starts at 0 mismatch), D (51 instead of 50, calculation error). Analyzing: (1) identify proportional relationship, (2) independent=time t, dependent=distance d, (3) verify table by plugging t into equation, (4) ensure includes t=0 for origin.

This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (snacks s we buy—we control), dependent is output/depends on independent (total cost c depends on snacks plus fixed ticket—results from independent). Equation: express dependent in terms of independent (c=12+2s: cost equals 12 plus 2 times snacks, dependent c on left, independent s in expression). For example, similar to movie with $10 ticket plus $3 drink, independent=drinks d, dependent=cost c, equation c=10+3d, table d:0,1,2 c:10,13,16, graph: x-drinks, y-cost, points (0,10),(1,13) line with y-intercept 10, slope 3. The correct identification is independent s (snacks), dependent c (total cost), as we choose snacks and cost results. Errors include reversing (cost independent, snacks dependent—backward) or misidentifying constants as variables. Analyzing: (1) identify relationship (cost depends on snacks with fixed $12 plus $2 each), (2) determine independent (s: we choose number) and dependent (c: result), (3) write equation (c=12+2s), (4) create table (s:0,1,2; c:12,14,16), (5) graph (x=s, y=c, plot pairs, line through (0,12)), (6) connect (equation matches table✓, slope=2✓). Axes convention: independent horizontal (x-axis: snacks), dependent vertical (y-axis: cost).