Represent Proportional Relationships by Equations - 7th Grade Math

1. What does the constant $k$ represent in the proportional equation $y = kx$?

   $k$ is the constant of proportionality (unit rate). It's the ratio $\frac{y}{x}$ that stays constant.

2. Identify the equation if 3 tickets cost $24$ dollars and total cost $t$ is proportional to tickets $n$.

   $t = 8n$. Find unit price: $\frac{24}{3} = 8$ dollars per ticket.

3. Identify the constant of proportionality in the equation $t = pn$ (between $t$ and $n$).

   $p$. The coefficient of $n$ is the constant of proportionality.

4. What is the constant of proportionality $k$ if a point $(x,y) = (5,20)$ is on $y = kx$?

   $k = 4$. Substitute $(5,20)$ into $y = kx$: $20 = k(5)$, so $k = 4$.

5. What is $y$ if $y$ is proportional to $x$ with $k = 7$ and $x = 3$?

   $y = 21$. Substitute: $y = 7(3) = 21$.

6. What is the unit rate (constant of proportionality) if $\frac{y}{x} = 7$ for all $x \ne 0$?

   $k = 7$. Since $\frac{y}{x} = k$ in proportional relationships.

7. What is the number of items $n$ if $t = 56$ dollars and $p = 8$ dollars per item?

   $n = 7$. Solve $56 = 8n$: $n = \frac{56}{8} = 7$.

8. What equation represents a runner traveling $150$ meters in $30$ seconds at constant speed, with $d$ vs. $t$?

   $d = 5t$. Speed is $\frac{150}{30} = 5$ meters per second.

9. Write an equation for total cost $t$ when price per item is $p = 6$ dollars and number of items is $n$.

   $t = 6n$. Substitute $p = 6$ into the formula $t = pn$.

10. Find $p$ if $t = 36$, $n = 8$, and $t = pn$.

    $p = \frac{9}{2}$. Solve $36 = p(8)$ by dividing both sides by $8$.

11. What is $y$ when $y = \frac{3}{5}x$ and $x = 20$?

    $y = 12$. Calculate $y = \frac{3}{5}(20) = \frac{60}{5} = 12$.

12. Identify the error: "$y = k + x$ represents a proportional relationship." What is the correct form?

    Correct form: $y = kx$. Proportional relationships have no constant term, only $y = kx$.

13. What equation represents $y$ proportional to $x$ with constant of proportionality $k = \frac{5}{2}$?

    $y = \frac{5}{2}x$. Substitute $k = \frac{5}{2}$ into $y = kx$.

14. What is the equation for earnings $E$ proportional to hours $h$ at hourly rate $r$?

    $E = rh$. Earnings equal hourly rate times hours worked.

15. Which equation shows a proportional relationship: $y = 3x$ or $y = 3x + 2$?

    $y = 3x$. Only $y = 3x$ passes through origin; $y = 3x + 2$ has y-intercept.

16. What must be true about the graph of a proportional relationship $y = kx$?

    It is a straight line through $(0,0)$. Proportional relationships always pass through the origin.

17. What is the equation for distance $d$ proportional to time $t$ at speed $v$?

    $d = vt$. Distance equals rate times time in proportional motion.

18. What is the equation for total cost $t$ when price per item is $p$ and number of items is $n$?

    $t = pn$. Total equals price per item times number of items.

19. Identify the constant of proportionality $k$ if the relationship is $y = -2x$.

    $k = -2$. The coefficient of $x$ in $y = -2x$ is $k = -2$.

20. Find $n$ if $t = 45$, $p = 9$, and $t = pn$.

    $n = 5$. Solve $45 = 9n$ by dividing both sides by $9$.

21. What does the constant of proportionality $k$ equal in the equation $y = kx$?

    $k = \frac{y}{x}$. Rearrange $y = kx$ by dividing both sides by $x$.

22. What is $x$ when $y = \frac{3}{4}x$ and $y = 15$?

    $x = 20$. Solve $15 = \frac{3}{4}x$: $x = 15 \times \frac{4}{3} = 20$.

23. What is the equation for distance $d$ traveled at constant speed $r$ for time $t$?

    $d = rt$. Distance equals rate times time for constant speed.

24. Which equation represents "total cost $t$ equals price per item $p$ times number of items $n$"?

    $t = pn$. Total equals rate times quantity in proportional relationships.

25. Identify $k$ in the equation $y = \frac{2}{3}x$.

    $k = \frac{2}{3}$. The coefficient of $x$ is the constant of proportionality.

26. Identify the constant of proportionality in the equation $t = pn$ when $t$ depends on $n$.

    $p$. The coefficient of $n$ in $t = pn$ is the constant $p$.

27. Identify $k$ in the equation $y = 12x$.

    $k = 12$. The coefficient of $x$ is the constant of proportionality.

28. What equation represents a proportional relationship with constant of proportionality $k$ between $y$ and $x$?

    $y = kx$. Direct proportionality means $y$ varies directly with $x$ by factor $k$.

29. Write the proportional equation if $y$ increases by $9$ when $x$ increases by $3$ and the relationship is proportional.

    $y = 3x$. Rate of change is $\frac{9}{3} = 3$, so $y = 3x$.

30. Find and correct the equation for proportionality: $y = \frac{x}{4}$ is written as $y = 4x$.

    Correct: $y = \frac{1}{4}x$. $y = \frac{x}{4}$ means $y = \frac{1}{4}x$, not $y = 4x$.

31. Find $t$ if $t = 4n$ and $n = 7$.

    $t = 28$. Substitute $n = 7$ into $t = 4n$: $t = 4(7) = 28$.

32. Write an equation for total cost $t$ when price per item is $p = \frac{5}{2}$ and number of items is $n$.

    $t = \frac{5}{2}n$. Substitute $p = \frac{5}{2}$ into the formula $t = pn$.

33. What is the total cost $t$ if $p = 4$ dollars per item and $n = 9$ items?

    $t = 36$. Calculate $t = 4 \times 9 = 36$.

34. What equation represents $y$ proportional to $x$ with constant of proportionality $k = \frac{3}{5}$?

    $y = \frac{3}{5}x$. Substitute $k = \frac{3}{5}$ into the general form $y = kx$.

35. Which point must lie on the graph of any proportional relationship between $y$ and $x$?

    $(0,0)$. When $x = 0$, $y = k(0) = 0$ for any value of $k$.

36. What is the general equation for a proportional relationship between $y$ and $x$?

    $y = kx$. Where $k$ is the constant ratio between $y$ and $x$.

37. Which equation shows $y$ proportional to $x$: $y = 3x + 2$ or $y = 3x$?

    $y = 3x$. Only $y = 3x$ passes through $(0,0)$; $y = 3x + 2$ doesn't.

38. What equation represents $y$ proportional to $x$ with constant of proportionality $k = 4$?

    $y = 4x$. Substitute $k = 4$ into the general form $y = kx$.

39. What is $k$ if $y = 18$ when $x = 6$ in a proportional relationship $y = kx$?

    $k = 3$. Solve $18 = k(6)$: $k = \frac{18}{6} = 3$.

40. What is the unit price $p$ if total cost is $t = 45$ dollars for $n = 9$ items?

    $p = 5$. Solve $45 = p(9)$: $p = \frac{45}{9} = 5$.

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