To solve this problem, we will follow these steps:

1. Represent the problem as a system of equations.
2. Solve the system of equations to find the exact costs of a juice box ($j$) and a bottle of water ($w$).
3. Justify Kara's claim that the juice box might cost $0.52 and the bottle of water might cost $0.33.

Step 1: Create the system of equations

Let:

• $j$ represent the cost of a juice box.
• $w$ represent the cost of a bottle of water.

From the information given:

• The first teacher spent $19.92 on 18 juice boxes and 32 bottles of water:
  $18j + 32w = 19.92$
• The second teacher spent $15.76 on 14 juice boxes and 26 bottles of water:
  $14j + 26w = 15.76$

The system of equations is:

1. & \quad 18j + 32w = 19.92 \\ 2. & \quad 14j + 26w = 15.76 \end{aligned}$$ ### Step 2: Solve the system of equations To solve, use substitution or elimination. Here, we'll use the **elimination method**. #### a) Eliminate one variable To eliminate $$j$$, we align the coefficients of $$j$$ by multiplying both equations: Multiply the first equation by 7 (to get $$126j$$): $$126j + 224w = 139.44$$ Multiply the second equation by 9 (to get $$126j$$): $$126j + 234w = 141.84$$ Now subtract the first equation from the second: $$(126j + 234w) - (126j + 224w) = 141.84 - 139.44$$ $$10w = 2.40$$ $$w = 0.24$$ #### b) Substitute $$w = 0.24$$ into one of the original equations Substitute $$w = 0.24$$ into the first equation ($$18j + 32w = 19.92$$): $$18j + 32(0.24) = 19.92$$ $$18j + 7.68 = 19.92$$ $$18j = 12.24$$ $$j = 0.68$$ ### Step 3: Verify Kara's claim Kara claimed $$j = 0.52$$ and $$w = 0.33$$. Substitute these values into both original equations to check: 1. For $$18j + 32w = 19.92$$: $$18(0.52) + 32(0.33) = 9.36 + 10.56 = 19.92 \quad \text{(True)}$$ 2. For $$14j + 26w = 15.76$$: $$14(0.52) + 26(0.33) = 7.28 + 8.58 = 15.76 \quad \text{(True)}$$ This shows Kara's claim is **correct**: these values satisfy the equations. ### Final Answer The actual cost of a juice box is **\$0.68**, and the cost of a bottle of water is **\$0.24**. --- Would you like further details or have any questions? ### Related Questions: 1. How can we verify solutions to any system of equations? 2. What are the key differences between substitution and elimination methods? 3. Why is it important to test solutions back into the original equations? 4. Could graphing the equations provide a visual solution to the problem? 5. How do rounding errors affect real-world problems like this one? **Tip:** Always double-check your results by substituting them back into the original equations!