To solve the problem, we will define variables for the costs of the items and set up equations based on the information given.
Let:
- \( x \) = cost of 1 chocolate
- \( y \) = cost of 1 candy
- \( z \) = cost of 1 bubblegum
From the problem, we have the following two equations based on the costs:
1. The cost of 3 chocolates, 2 candies, and 1 bubblegum is $14:
\[
3x + 2y + z = 14 \quad \text{(Equation 1)}
\]
2. The cost of 1 chocolate, 3 candies, and 5 bubblegums is also $14:
\[
x + 3y + 5z = 14 \quad \text{(Equation 2)}
\]
We need to find the total cost of 1 chocolate, 1 candy, and 1 bubblegum, which is represented as \( x + y + z \).
### Step 1: Solve the equations
First, we will manipulate these equations to eliminate one variable. Let's multiply Equation 1 by 5 to align the coefficients of \( z \):
\[
5(3x + 2y + z) = 5(14)
\]
\[
15x + 10y + 5z = 70 \quad \text{(Equation 3)}
\]
Now we will subtract Equation 2 from Equation 3:
\[
(15x + 10y + 5z) - (x + 3y + 5z) = 70 - 14
\]
\[
15x - x + 10y - 3y + 5z - 5z = 56
\]
\[
14x + 7y = 56
\]
### Step 2: Simplify the equation
Now, we can simplify this equation by dividing everything by 7:
\[
2x + y = 8 \quad \text{(Equation 4)}
\]
### Step 3: Solve for another variable
Next, we will manipulate Equation 2. We will multiply Equation 2 by 3:
\[
3(x + 3y + 5z) = 3(14)
\]
\[
3x + 9y + 15z = 42 \quad \text{(Equation 5)}
\]
Now, we will subtract Equation 1 from Equation 5:
\[
(3x + 9y + 15z) - (3x + 2y + z) = 42 - 14
\]
\[
3x - 3x + 9y - 2y + 15z - z = 28
\]
\[
7y + 14z = 28
\]
### Step 4: Simplify again
Now, divide everything by 7:
\[
y + 2z = 4 \quad \text{(Equation 6)}
\]
### Step 5: Solve for \( z \)
From Equation 6, we can express \( z \) in terms of \( y \):
\[
2z = 4 - y
\]
\[
z = 2 - \frac{y}{2}
\]
### Step 6: Substitute \( z \) back into Equation 4
Now substitute \( z \) back into Equation 4:
\[
2x + y = 8
\]
Substituting \( z \):
\[
2x + y = 8
\]
We can express \( x \) in terms of \( y \):
\[
x = 4 - \frac{y}{2}
\]
### Step 7: Find \( x + y + z \)
Now we can find \( x + y + z \):
\[
x + y + z = \left(4 - \frac{y}{2}\right) + y + \left(2 - \frac{y}{2}\right)
\]
Combining like terms:
\[
= 4 + 2 + y - \frac{y}{2} - \frac{y}{2}
\]
\[
= 6 + y - y = 6
\]
Thus, the cost of 1 chocolate, 1 candy, and 1 bubblegum is **$6**.
### Final Answer:
The cost of 1 chocolate, 1 candy, and 1 bubblegum is **$6**.
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