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The Johnsons and the Wilsons went to the Burger Barn for dinner.

The Johnsons bought 2 burger meals and 5 hot dog meals for $46.
The Wilsons bought 5 burger meals and 2 hot dog meals for $52.
When you solve, you get a solution of (8, 6).
(3 points)
What does the 8 mean in context?
What does the 6 mean in context?
Who got the better deal?
:: How many burger meals
:: The cost of a burger meal
:: The total cost of a burger + hot dog meal
:: How many hot dog meals
:: The cost of a hot dog meal
:: Neither
:: both
:: The Johnson's
:: The Wilsons

3 answers

To interpret the solution (8, 6) in the context of the given problem, we need to define what the variables represent.

Let's assume:

  • \( x \) represents the cost of a burger meal,
  • \( y \) represents the cost of a hot dog meal.

Given the equations:

  1. For the Johnsons: \( 2x + 5y = 46 \)
  2. For the Wilsons: \( 5x + 2y = 52 \)

After solving the equations, we get the solution \( (8, 6) \):

  • The 8 represents the cost of a burger meal. So, each burger meal costs $8.
  • The 6 represents the cost of a hot dog meal. So, each hot dog meal costs $6.

Now, to determine who got the better deal, we need to analyze the total costs for both families:

Johnsons:

  • They bought 2 burger meals and 5 hot dog meals.
  • Total cost: \( 2(8) + 5(6) = 16 + 30 = 46 \) (which fits the given information).

Wilsons:

  • They bought 5 burger meals and 2 hot dog meals.
  • Total cost: \( 5(8) + 2(6) = 40 + 12 = 52 \) (which also fits the given information).

Now, let's figure out how much each family spent per meal:

  • Total meals for Johnsons = 2 + 5 = 7 meals.

    • Cost per meal for Johnsons: \[ \frac{46}{7} \approx 6.57 \text{ per meal} \]

  • Total meals for Wilsons = 5 + 2 = 7 meals.

    • Cost per meal for Wilsons: \[ \frac{52}{7} \approx 7.43 \text{ per meal} \]

Comparing the average cost per meal:

  • Johnsons: approximately $6.57 per meal
  • Wilsons: approximately $7.43 per meal

Since the average cost per meal is lower for the Johnsons, we can conclude that the Johnsons got the better deal.

Final Answer:

  • The 8 means the cost of a burger meal ($8).
  • The 6 means the cost of a hot dog meal ($6).
  • The Johnsons got the better deal.
Both of these tables represent linear graphs:
*

-3
8
-2
10
-1
12

14
1
16
2
18
3
20
4
22

У
-3
-2

0

2

4

8
3
10
4
12
Does the line through the first set of points intersect with the line through the second set of points?
(1 point)
• Yes...these lines are not parallel nor perpendicular
• No...these lines are parallel
• Yes... these lines are parallel
• You cannot tell without a graph
• No... these lines are perpendicular
• Yes... these lines are perpendicular

To determine whether the lines represented by the two sets of points intersect, we need to first understand the slopes of the lines and their equations.

  1. First Set of Points:

    • Points: (-3, 8), (-2, 10), (-1, 12)

    • Slope Calculation:

      • Using the points (-2, 10) and (-1, 12): \[ \text{slope} = \frac{12 - 10}{-1 - (-2)} = \frac{2}{1} = 2 \]

    • The slope of the first line is 2.

  2. Second Set of Points:

    • Points: (-3, -2), (0, 2), (3, 10), (4, 12)

    • Slope Calculation:

      • Using the points (0, 2) and (3, 10): \[ \text{slope} = \frac{10 - 2}{3 - 0} = \frac{8}{3} \]

    • The slope of the second line is \(\frac{8}{3}\) (or approximately 2.67).

Conclusion:

  • The slopes of the two lines are different (2 and \(\frac{8}{3}\)).
  • Since the slopes are not equal, the lines are not parallel.
  • Additionally, they are not perpendicular, because the product of their slopes is not -1 (the product of 2 and \(\frac{8}{3}\) is \(\frac{16}{3}\), which is not -1).

Final Answer:

• Yes...these lines are not parallel nor perpendicular.