Analyze Dependent and Independent Variables - 6th Grade Math

A plant grows according to the equation $h = 15 + 2.5d$ where $d$ is days since planting and $h$ is height in centimeters. If a student wants to predict when the plant will reach 35 cm, which approach correctly identifies the variables for solving this problem?
Answer: Substitute $h = 35$ and solve for $d$ because height is given and days is unknown 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.
A science class warms water. The temperature starts at 18°C and increases by 4°C each minute. Let $t$ be time in minutes and $T$ be temperature in °C. Which statement correctly connects the equation, table, and graph?
Answer: Equation $T=4t+18$ generates the table values, and plotting the table as $(t,T)$ gives a straight line with y-intercept 18. 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), dependent is output/depends on independent (temperature T=4t+18). Equation: express dependent in terms of independent (T=4t+18). Table: list t:0,1,2, calculate T:18,22,26 from equation. Graph: independent on x-axis (t), dependent on y-axis (T), plot (t,T) with y-intercept 18, slope 4. Choice A correctly connects equation T=4t+18 to table and graph with intercept 18. Errors like B (reversed variables and axes), C (swapped coefficients, wrong origin), D (wrong plotting pairs and slope). Analyzing: (1) identify relationship with initial temperature, (2) independent=time t, dependent=T, (3) equation T=rate×t + initial, (4) table from equation, (5) graph (t,T) with intercept matching initial 18, (6) connect representations consistently.
Examine the coordinate plane shown. If $x$ represents the independent variable and $y$ represents the dependent variable, which statement best describes the relationship between the variables for the first 4 hours?
Answer: $y$ decreases at a constant rate of 2 units per hour, so $y = 8 - 2x$ Looking at the plotted points: (0,8), (1,6), (2,4), (3,2), (4,0), the $y$ -values decrease by 2 units for each 1-unit increase in $x$ . Starting at 8 when $x=0$ , the equation is $y = 8 - 2x$ . Choice A shows increase instead of decrease. Choice C has wrong rate and operation. Choice D has wrong operation sign.
A bike rental shop charges a $3 starting fee plus $2 for each hour you rent the bike. Which statement correctly identifies the independent and dependent variables?
Answer: Independent: hours rented; Dependent: total cost 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 rented h we choose), dependent is output/depends on independent (total cost c depends on hours plus starting fee). Equation: express dependent in terms of independent (c=2h+3: cost equals 2 times hours plus 3, 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=2×1+3=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 c=2h+3 graphs through (0,3). In this bike rental example, independent=hours rented (we control how long we rent), dependent=total cost (results from hours plus fee), so choice B correctly identifies them. Common errors include reversing variables (like A, making cost independent, which is backward) or confusing constants as variables (like C or D). To analyze: (1) identify relationship (cost depends on hours rented with fixed fee), (2) determine independent (hours: we choose) and dependent (cost: result), (3) avoid mistaking fees or rates as variables.
A movie theater charges a $12 ticket plus $2 per snack. Let $s$ be the number of snacks and let $c$ be the total cost in dollars. Which choice correctly identifies the independent and dependent variables?
Answer: Independent: $s$ (snacks); Dependent: $c$ (total cost) 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).
A runner jogs at a constant speed of 6 miles per hour. Let $t$ be time in hours and let $d$ be distance in miles. Which equation shows $d$ as a function of $t$ ?
Answer: $d=6t$ 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).
A car travels at a constant speed of 50 miles per hour. Let $t$ be time in hours and $d$ be distance in miles. Which table correctly shows values that satisfy $d = 50t$ ?
Answer: $t: 0,1,2,3$ and $d: 0,50,100,150$ This question is all about using a rule to fill in a table. The rule here is $d = 50t$ , which means "distance equals 50 times the time." To find each distance, just multiply the time $t$ by 50: - When $t = 0$ : $d = 50 \times 0 = 0$ - When $t = 1$ : $d = 50 \times 1 = 50$ - When $t = 2$ : $d = 50 \times 2 = 100$ - When $t = 3$ : $d = 50 \times 3 = 150$ So the table is $t: 0,1,2,3$ and $d: 0,50,100,150$ . Think of it like steps: every hour you "step" forward 50 miles. At zero hours, you haven't moved yet, so $d = 0$ . Try this at home: pick your own speed, like 30 miles per hour. Make a table for $t = 0,1,2,3$ and multiply each time by 30. You've got it!
A taxi charges a $4 start fee plus $2 per mile. Let $m$ be miles traveled and $c$ be total cost in dollars. Which equation represents the relationship?
Answer: $c=2m+4$ 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 (miles m traveled—we control distance), dependent is output/depends on independent (cost c depends on miles plus fixed fee—results from independent). Equation: express dependent in terms of independent (c=4+2m: cost equals 4 plus 2 times miles, dependent c on left, independent m in expression). For example, similar to cab with $3 fee plus $1.5/mile, independent=miles m, dependent=cost c, equation c=3+1.5m, table m:0,1,2 c:3,4.5,6, graph: x-miles, y-cost, points (0,3),(1,4.5) line y-intercept 3, slope 1.5. The correct equation is c=2m+4 (or equivalently c=4+2m), matching fixed $4 plus $2 per mile. Errors include reversing variables in equation (like m=2c+4) or wrong coefficients (c=4m+2 switching rates). Analyzing: (1) identify relationship (cost depends on miles with start fee and per-mile rate), (2) determine independent (m) and dependent (c), (3) write equation (c=4+2m), (4) create table (m:0,1,2; c:4,6,8), (5) graph (x=m, y=c, plot pairs, line through (0,4)), (6) connect (equation matches table: 4+2*1=6✓, slope=2✓). Axes convention: independent horizontal (x-axis: miles), dependent vertical (y-axis: cost).
Based on the table shown, which equation best represents the relationship between the number of workers $w$ and the time $t$ (in hours) needed to complete a job?
Answer: $t = \frac{24}{w}$ where $w$ is independent and $t$ is dependent From the table, as workers increase, time decreases in an inverse relationship: $2 \times 12 = 24$ , $3 \times 8 = 24$ , $4 \times 6 = 24$ . The number of workers is chosen first (independent), determining the time needed (dependent). The equation is $t = \frac{24}{w}$ . Choice B shows direct variation. Choices C and D incorrectly identify the independent variable.
A plant is 8 cm tall when it is first measured and grows 3 cm each week. Let $w$ be the number of weeks and let $h$ be the height (cm). Which equation represents the relationship?
Answer: $h = 3w + 8$ 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 (weeks w passed, time we wait—we control), dependent is output/depends on independent (height h depends on weeks, result depends on time—results from independent). Equation: express dependent in terms of independent (h=8+3w: height equals 8 plus 3 times weeks, dependent h on left, independent w in expression). Table: list independent values (w: 0,1,2,3), calculate dependent using equation (if w=1, h=8+3×1=11; w=2, h=14; etc.). Graph: independent on x-axis (horizontal: weeks), dependent on y-axis (vertical: height), plot ordered pairs ((1,11), (2,14),...), line with intercept. In this plant growth example, independent=weeks w, dependent=height h, equation h=3w+8, table w:0,1,2 h:8,11,14, graph x=weeks, y=height, line with slope 3 and y-intercept 8. The correct equation is h=3w+8 as in choice A; errors like C reverse variables, D misapplies parentheses, B swaps coefficients.
A car rental company charges $30 per day plus $0.25 per mile driven. Two students write different equations: Student A writes $C = 30d + 0.25m$ and Student B writes $C = 30 + 0.25m$ (for one day). In terms of independent and dependent variables, which analysis is most accurate?
Answer: Both students correctly identify cost as dependent, but Student A's equation handles multiple independent variables 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.
A movie theater charges a $5 admission fee plus $3 per hour for each movie ticket. If $h$ represents the number of hours a movie lasts and $C$ represents the total cost of a ticket, which statement correctly identifies the variables and their relationship?
Answer: $h$ is independent, $C$ is dependent, and $C = 3h + 5$ 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.
A plant is 8 cm tall when it is measured at week 0, and it grows 3 cm each week. Let $w$ be the number of weeks since week 0, and let $h$ be the height in centimeters. Which table matches the relationship between $w$ and $h$ ?
Answer: $w: 0,1,2,3$ and $h: 8,11,14,17$ 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 (weeks w since week 0—we observe over time), dependent is output/depends on independent (height h depends on weeks—results from independent). Equation: express dependent in terms of independent (h=8+3w: height equals 8 plus 3 times weeks, dependent h on left, independent w in expression). Table: list independent values (w: 0,1,2,3), calculate dependent using equation (if w=0, h=8; w=1, h=11; etc.). Graph: independent on x-axis (horizontal: weeks), dependent on y-axis (vertical: height), plot ordered pairs ((0,8), (1,11),...), line through (0,8) with slope 3. For example, similar to plant growth starting at 5 cm growing 2 cm/week, independent=weeks w, dependent=height h, equation h=5+2w, table w:0,1,2,3 h:5,7,9,11, graph: x-weeks, y-height, points (0,5),(1,7) forming line with y-intercept 5 and slope 2. The correct table is w:0,1,2,3 and h:8,11,14,17, matching initial 8 cm plus 3 cm per week. Errors include wrong calculations (like h:8,10,12,14 assuming +2 instead of +3) or starting w at 1 instead of 0. Analyzing: (1) identify relationship (height depends on weeks with initial height and growth rate), (2) determine independent (w) and dependent (h), (3) write equation (h=8+3w), (4) create table (w:0,1,2,3; h=8+3*0=8, etc.), (5) graph (x=w, y=h, plot pairs), (6) connect (table matches equation: 8+3*1=11✓). Axes convention: independent horizontal (x-axis: weeks), dependent vertical (y-axis: height).
A school store sells notebooks for $3 each. Let $n$ be the number of notebooks bought and let $c$ be the total cost (in dollars). Which statement correctly identifies the independent and dependent variables and gives an equation for the relationship?
Answer: Independent: $n$ ; Dependent: $c$ ; Equation: $c=3n$ 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 (number of notebooks n we buy—we control), dependent is output/depends on independent (cost c depends on notebooks—results from independent). Equation: express dependent in terms of independent (c=3n: cost equals 3 times notebooks, dependent c on left, independent n in expression). For example, in a similar cost scenario like buying apples at $2 each, independent=number a (apples bought), dependent=cost c (dollars spent), equation c=2a, table a:0,1,2,3 c:0,2,4,6 (each from 2×a), graph: x-axis apples, y-axis cost, points (0,0),(1,2),(2,4),(3,6) forming line through (0,0) with slope 2 matching equation coefficient. The correct choice identifies independent as n (notebooks), dependent as c (cost), with equation c=3n, as we choose notebooks and cost results proportionally. Errors include reversing variables (cost independent, notebooks dependent—backward), or wrong equation (additive like c=n+3 when should be multiplicative c=3n). To analyze: (1) identify relationship (cost depends on notebooks at $3 each), (2) determine independent (n: we choose number bought) and dependent (c: result of buying that many), (3) write equation (c=3n), (4) connect to table or graph (e.g., for n=1, c=3; n=2, c=6, plotting (1,3),(2,6) with slope 3). Axes convention: independent horizontal (x-axis: notebooks), dependent vertical (y-axis: cost).
A runner moves at a constant speed of 7 meters per second. Let $t$ be time in seconds and $d$ be distance in meters. Which ordered pairs should be graphed for the relationship $d = 7t$ (with $t$ on the x-axis and $d$ on the y-axis)?
Answer: $(0,0), (1,7), (2,14), (3,21)$ 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 in seconds, periods we measure—we control), dependent is output/depends on independent (distance d depends on time, result depends on period—results from independent). Equation: express dependent in terms of independent (d=7t: distance equals 7 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=7×1=7; t=2, d=14; etc.). Graph: independent on x-axis (horizontal: time), dependent on y-axis (vertical: distance), plot ordered pairs ((1,7), (2,14),...), proportional through origin. In this runner example, independent=time t, dependent=distance d, equation d=7t, table t:0,1,2,3 d:0,7,14,21, graph x=time, y=distance, points (0,0),(1,7),(2,14),(3,21) line with slope 7. The correct ordered pairs are (0,0),(1,7),(2,14),(3,21) as in C (t on x, d on y); errors like A reverse coordinates, B starts at (0,7), D mixes values.
A plant is 8 cm tall when measured at week 0 and grows 3 cm each week. Let $w$ be the number of weeks and $h$ be the height in cm. Which equation correctly relates the variables?
Answer: $h = 3w + 8$ 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 (weeks w we measure), dependent is output/depends on independent (height h depends on weeks plus initial height). Equation: express dependent in terms of independent (h=3w+8: height equals 3 times weeks plus 8, dependent h on left, independent w in expression). Table: list independent values (w: 0,1,2,3), calculate dependent using equation (if w=0, h=8; w=1, h=11; etc.). Graph: independent on x-axis (horizontal: weeks), dependent on y-axis (vertical: height), plot ordered pairs ((0,8), (1,11),...), line with y-intercept 8. In this plant growth example, choice A correctly relates h=3w+8, with growth rate 3 and initial 8. Errors like B (h=8w+3, swapping coefficients), C (solving for w instead), or D (distributive form incorrect). Analyzing: (1) identify relationship (height depends on weeks with initial value), (2) independent=weeks w, dependent=height h, (3) equation h=rate×w + initial, (4) check at w=0, h=8 matches.
Refer to the graph showing the relationship between study time and test scores. A student claims that $s = 40 + 8h$ represents this relationship, where $h$ is hours studied and $s$ is the test score. What error exists in the student's analysis?
Answer: The equation is correct, but the slope should be 5 points per hour, not 8 Hours studied $h$ is correctly independent (student chooses study time), and test score $s$ is correctly dependent. However, from the graph points (0,40), (2,50), (4,60), (6,70), the slope is $\frac{50-40}{2-0} = 5$ points per hour, not 8. The correct equation is $s = 40 + 5h$ . The student used the wrong slope coefficient.
A delivery truck travels at 45 mph on a highway. The relationship between time $t$ (in hours) and distance $d$ (in miles) is shown in the graph. Which statement correctly analyzes both the mathematical relationship and the real-world constraints?
Answer: $d = 45t$ with $t$ independent; realistic values are $t ≥ 0$ since negative time is meaningless Time is the independent variable (you choose how long to drive), distance is dependent. The equation $d = 45t$ shows distance equals rate times time. In real-world context, $t ≥ 0$ because negative time doesn't make sense for actual travel. Choice A incorrectly accepts negative time. Choices C and D incorrectly identify distance as independent.
A streaming service costs $12 per month plus a one-time sign-up fee of $5. Let $m$ be the number of months and $c$ be the total cost. Which choice correctly identifies the independent variable and writes an equation for $c$ in terms of $m$ ?
Answer: Independent: $m$ ; equation: $c = 12m + 5$ 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 (months m we subscribe), dependent is output/depends on independent (total cost c depends on months plus fee). Equation: express dependent in terms of independent (c=12m+5: cost equals 12 times months plus 5, dependent c on left, independent m in expression). Table: list independent values (m: 0,1,2), calculate dependent (m=1, c=17). Graph: independent on x-axis (horizontal: months), dependent on y-axis (vertical: cost), plot with y-intercept 5. Choice B correctly identifies independent m and equation c=12m+5. Errors like A (independent c, wrong; equation solved for m), C (coefficients swapped), D (independent as constant 12, wrong equation). Analyzing: (1) identify relationship with fixed fee, (2) independent=months m, dependent=cost c, (3) equation c=rate×m + fee, (4) ensure independent is input variable.
A gym membership costs $10 to sign up plus $5 each month. Let $m$ be the number of months and $C$ be the total cost. Which description correctly connects the equation, table, and graph for this situation?
Answer: Equation $C=5m+10$ ; table starts at $C=10$ when $m=0$ and increases by 5 each month; graph is a line with y-intercept 10. 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 (months m subscribed, periods we track—we control), dependent is output/depends on independent (cost C depends on months, result depends on period—results from independent). Equation: express dependent in terms of independent (C=10+5m: cost equals 10 plus 5 times months, dependent C on left, independent m in expression). Table: list independent values (m: 0,1,2,3), calculate dependent using equation (if m=1, C=10+5×1=15; m=2, C=20; etc.). Graph: independent on x-axis (horizontal: months), dependent on y-axis (vertical: cost), plot ordered pairs ((1,15), (2,20),...), line with y-intercept 10. For this gym membership, independent=months m, dependent=cost C, equation C=5m+10, table m:0,1,2 C:10,15,20 increasing by 5, graph line with slope 5 and y-intercept 10. The correct description connects equation C=5m+10, table starting at 10 and increasing by 5, graph with y-intercept 10 as in C; errors like A omit intercept, B/D swap or misstate values.

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