To solve the problem, we will match the representations provided in the image (graphs, tables, and word problems) and group those that describe the same relationship. Here's a breakdown:

Matching Representations

1. Graph A:

   • The line passes through $(0, 0)$, with a slope of $2$ (rise of $2$, run of $1$).
   • Equation: $y = 2x$.

2. Graph B:

   • Points: $(0, 1), (1, 4), (2, 7), (3, 10)$.
   • This suggests a linear equation with a slope of $3$: $y = 3x + 1$.

3. Graph C:

   • Points: $(0, 0), (1, 2), (2, 4), (3, 6)$.
   • This aligns with a linear equation $y = 2x$, the same as Graph A.

4. Graph D:

   • The line passes through $(0, 5)$ and has a slope of $1$ (rise = run).
   • Equation: $y = x + 5$.

5. Scenario E:

   • Describes a decrease of $0.5 \%$ (a constant rate of change) per hour. However, the specific values are not clearly shown, making it less directly comparable to others.

6. Scenario F:

   • The temperature starts at $0^\circ C$ and increases by $2^\circ C$ every hour.
   • Equation: $y = 2x$. Matches Graph A and Graph C.

7. Table G:

   • $x = 0, 2, 4, 6$; $y = 10, 9, 8, 7$.
   • Shows a constant decrease ($-1$ per unit $x$).
   • Equation: $y = -x + 10$.

8. Table H:

   • $x = 0, 1, 2, 3, 4$; $y = 1, 4, 7, 10, 13$.
   • A constant rate of increase ($+3$ per unit $x$).
   • Equation: $y = 3x + 1$. Matches Graph B.

9. Scenario I:

   • Starting with $1$ bacterium, the population increases by $3$ every minute.
   • Equation: $y = 3x + 1$. Matches Graph B and Table H.

10. Graph J:

    • A line passing through $(0, 10)$ with a slope of $-1$.
    • Equation: $y = -x + 10$. Matches Table G.

11. Table K:

    • Points: $(0, 10), (4, 8), (6, 7), (8, 6)$.
    • Matches $y = -\frac{1}{2}x + 10$, unique from others.

Final Grouping of Representations

• Group 1: $y = 2x$

  • Graph A, Graph C, Scenario F.

• Group 2: $y = 3x + 1$

  • Graph B, Table H, Scenario I.

• Group 3: $y = -x + 10$

  • Table G, Graph J.

• Unique:

  • Graph D ($y = x + 5$), Table K ($y = -\frac{1}{2}x + 10$).

Would you like further details or explanations about any specific part of this matching?

5 Questions for Practice

1. What is the slope and y-intercept of $y = -x + 10$?
2. How do you find the equation of a line from a graph?
3. What is the relationship between a table of values and the slope of a line?
4. How does a graph with a negative slope differ from one with a positive slope?
5. How do linear relationships relate to real-world scenarios?

Tip:

Always identify the slope and intercept first when comparing relationships!