Analyze Dependent and Independent Variables - 6th Grade Math

1. What is the equation for total cost with a $5$ dollar fee plus $2$ dollars per item $n$?

   $C = 2n + 5$. Fixed fee plus variable cost per item.

2. What is the independent variable in $d = 65t$, where $d$ is distance and $t$ is time?

   $t$. Time is chosen first; distance depends on it.

3. Identify the equation that matches this table: $x: 0,1,2$ and $y: 3,5,7$.

   $y = 2x + 3$. Each $x$ increase of $1$ gives $y$ increase of $2$, starting at $3$.

4. What is the equation for total cost $C$ with a $5$ dollar start fee plus $2$ dollars per mile $m$?

   $C = 2m + 5$. Fixed fee plus variable cost per mile.

5. What is the independent variable in $d = 65t$ if $d$ is distance and $t$ is time?

   $t$. Time is controlled/chosen; distance depends on it.

6. What does it mean for a variable to be dependent in a relationship between two quantities?

   It is the output that changes based on the other variable. Its value is determined by the independent variable.

7. What is $t$ when $d = 260$ in the equation $d = 65t$?

   $4$. Solve $260 = 65t$ by dividing both sides by $65$.

8. In $F = \frac{9}{5}C + 32$, which variable is independent if you choose the Celsius value first?

   $C$. You input Celsius; Fahrenheit is calculated from it.

9. What is the dependent variable in $d = 65t$ if $d$ is distance and $t$ is time?

   $d$. Distance depends on time; it changes based on how long you travel.

10. What is the constant of proportionality in the equation $d = 65t$?

    $65$. The coefficient of $t$ represents the rate of change.

11. Which equation matches: "A taxi charges a $5$ dollar start fee plus $2$ dollars per mile $m$"?

    $C = 5 + 2m$. Fixed fee plus variable rate per mile gives total cost.

12. Which equation expresses $y$ as a function of $x$ for a constant rate of $3$ and start value $-2$?

    $y = 3x - 2$. Linear equation with slope $3$ and y-intercept $-2$.

13. What does the slope mean in $d = 65t$ when $d$ is miles and $t$ is hours?

    $65$ miles per hour. The coefficient shows the constant speed.

14. What is the slope of the line for $d = 65t$ when graphing $d$ versus $t$?

    $65$. Slope equals the coefficient of the independent variable.

15. What is the equation for distance $d$ after $t$ hours at a constant speed of $65$ miles per hour?

    $d = 65t$. Distance equals speed times time.

16. What is the dependent variable in $d = 65t$, where $d$ is distance and $t$ is time?

    $d$. Distance depends on the time traveled.

17. What is the y-intercept of the relationship $d = 65t$ in a $t$-$d$ graph?

    $(0, 0)$. When $t = 0$, $d = 65(0) = 0$.

18. What is the $d$-intercept of $d = 65t$ (the value of $d$ when $t = 0$)?

    $0$. When $t=0$, $d=65(0)=0$.

19. What ordered pair $(t,d)$ matches $d = 65t$ when $t = 2$?

    $(2,130)$. Substitute $t=2$: $d=65(2)=130$.

20. What is $d$ when $t = 3$ in the equation $d = 65t$?

    $195$. Substitute $t = 3$: $d = 65(3) = 195$.

21. What is the slope (rate of change) in $C = 5 + 2m$ when graphed as $(m, C)$?

    $2$. Cost increases by $2 for each additional mile.

22. What is the equation for a starting height of $50$ cm that increases by $3$ cm per day $d$?

    $h = 50 + 3d$. Initial height plus growth rate times days.

23. What does the negative slope mean in $M = 20 - 3d$ (money left vs. days)?

    $M$ decreases by $3$ each time $d$ increases by $1$. Negative slope shows money decreases over time.

24. Which point is on the graph of $y = 2x + 1$: $(2,5)$ or $(2,6)$?

    $(2, 5)$. Check: $y = 2(2) + 1 = 5$ ✓.

25. What is the y-intercept in $C = 5 + 2m$ when graphed as $(m, C)$?

    $5$. When $m = 0$, $C = 5 + 2(0) = 5$ (initial fee).

26. What does the coefficient $65$ represent in $d = 65t$ (distance vs. time at constant speed)?

    Speed is $65$ distance-units per $1$ time-unit. The constant rate shows distance covered per unit time.

27. Identify the correct table rule: if $(t, d)$ follows $d = 4t$, what is $d$ when $t = 7$?

    $28$. Multiply: $d = 4(7) = 28$.

28. What is the dependent variable in an equation written as $y$ in terms of $x$?

    $y$. The dependent variable's value depends on the independent variable.

29. What is $d$ when $d = 65t$ and $t = 2$?

    $d = 130$. Substitute $t = 2$: $d = 65(2) = 130$.

30. What is $M$ when $d = 4$ for the equation $M = 20 - 3d$?

    $8$. Substitute: $M = 20 - 3(4) = 20 - 12 = 8$.

31. What is the rate of change between consecutive points in the table $x: 0,1,2$ and $y: 3,5,7$?

    $2$. Change in $y$ divided by change in $x$: $\frac{5-3}{1-0}=2$.

32. Identify the equation for: "You have $20$ dollars and spend $3$ dollars per day $d$".

    $M = 20 - 3d$. Start amount minus spending rate times days.

33. What is the meaning of the y-intercept in $y = 3x - 2$ when $x$ is the independent variable?

    When $x = 0$, $y = -2$. Y-intercept is the value of $y$ when input $x$ equals zero.

34. In $C = 2m + 5$, which variable is dependent if $m$ is miles and $C$ is total cost?

    $C$. Cost depends on miles traveled.

35. What is $h$ after $7$ days if $h = 50 + 3d$?

    $h = 71$. Substitute $d = 7$: $h = 50 + 3(7) = 71$.

36. Which variable is dependent in $C = 5 + 2m$ if $m$ is miles and $C$ is total cost?

    $C$. Cost depends on miles driven; it's the output variable.

37. What is $C$ when $m = 4$ in the relationship $C = 2m + 5$?

    $13$. Substitute: $C=2(4)+5=8+5=13$.

38. What is $t$ when $d = 65t$ and $d = 195$?

    $t = 3$. Solve $195 = 65t$ by dividing both sides by $65$.

39. What is the correct ordered pair for $t = 4$ in the relationship $d = 65t$?

    $(4, 260)$. Substitute $t = 4$: $d = 65(4) = 260$.

40. Identify the equation that matches the table $(x,y)$: $(0,1)$, $(1,3)$, $(2,5)$, $(3,7)$.

    $y = 2x + 1$. Pattern shows $y$ increases by $2$ for each unit increase in $x$.

41. Which variable is independent in $C = 2n + 5$: $C$ or $n$?

    $n$. Number of items is chosen; cost depends on it.

42. What is the independent variable in $h = 50 + 3d$?

    $d$. Days are chosen; height depends on days.

43. What is $F$ when $C = 0$ in $F = \frac{9}{5}C + 32$?

    $32$. Water freezes at $0°C$ and $32°F$.

44. Which equation represents a line with slope $4$ and $y$-intercept $-1$?

    $y = 4x - 1$. Slope-intercept form: $y=mx+b$ where $m=4$, $b=-1$.

45. What is the slope (rate of change) of the relationship $d = 65t$?

    $65$. Coefficient of $t$ represents miles per hour.

46. What is $C$ when $m = 6$ for the relationship $C = 5 + 2m$?

    $17$. Substitute: $C = 5 + 2(6) = 5 + 12 = 17$.

47. What is the independent variable in an equation written as $y$ in terms of $x$?

    $x$. The independent variable is chosen freely and determines the dependent variable.

48. What does it mean for a variable to be independent in a relationship between two quantities?

    It is the input you choose; the other variable depends on it. You control this value; it determines the other.

49. What ordered pair form matches "input, output" for independent and dependent variables?

    $(x, y)$. Independent variable comes first, dependent variable second in ordered pairs.

50. What is $C$ when $C = 2n + 5$ and $n = 6$?

    $C = 17$. Substitute $n = 6$: $C = 2(6) + 5 = 17$.

51. What does it mean to write one quantity "in terms of" another quantity?

    Express one variable using the other variable. Write an equation showing how one variable is calculated from the other.

52. What is the equation for distance at constant speed $65$ mph after $t$ hours?

    $d = 65t$. Distance equals speed times time for constant motion.

53. What is the height $h$ after $5$ weeks in $h = 10 + 2w$?

    $20$. Substitute: $h = 10 + 2(5) = 10 + 10 = 20$.

54. What is the value of $d$ when $M = 0$ in $M = 20 - 3d$?

    $\frac{20}{3}$. Solve $0 = 20 - 3d$: $3d = 20$, so $d = \frac{20}{3}$.

55. Identify the ordered pair format when graphing distance $d$ versus time $t$.

    Plot $(t, d)$. Independent variable first, dependent second in ordered pairs.

56. Which equation matches: "A plant is $10$ cm tall and grows $2$ cm per week $w$"?

    $h = 10 + 2w$. Initial height plus growth rate times weeks.

57. Which ordered pair $(t,d)$ is on the graph of $d = 65t$: $(3,195)$ or $(3,190)$?

    $(3,195)$. Check: $65(3)=195$, not $190$.

58. Which variable depends on time in the equation $d = 65t$: $d$ or $t$?

    $d$. Distance changes based on time, making it dependent.

59. What ordered pair $(m,C)$ matches $C = 2m + 5$ when $m = 0$?

    $(0,5)$. At $0$ miles, only the $5$ start fee applies.

Study these flashcards