Create a system of equations for the following problem and solve.

Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than half of her groceries, and that your other roommate’s groceries were $2.10 more than your groceries. How much was each of your share of the groceries?



To solve this problem, we'll define variables for the cost of each person's groceries and set up a system of equations based on the information provided.

### Let:
- $ x $ be the cost of your groceries,
- $ y $ be the cost of Sarah's groceries,
- $ z $ be the cost of your other roommate's groceries.

### The problem gives us the following information:
1. The total bill for all groceries is $82:
   \[
   x + y + z = 82
   \]
2. Your groceries were $0.05 cheaper than half of Sarah's groceries:
   \[
   x = \frac{1}{2}y - 0.05
   \]
3. Your other roommate's groceries were $2.10 more than your groceries:
   \[
   z = x + 2.10
   \]

Now, we'll solve this system of equations.

### Step 1: Write down the system of equations.
\[
\begin{align*}
x + y + z &= 82 \quad \text{(Equation 1)} \\
x &= \frac{1}{2}y - 0.05 \quad \text{(Equation 2)} \\
z = x + 2.10 \quad \text{(Equation 3)}
\end{align*}
\]

### Step 2: Substitute Equation 2 and Equation 3 into Equation 1.

First, solve for $ z $ in terms of $ y $:
- From Equation 2:
  \[
  x = \frac{1}{2}y - 0.05
  \]
- Substitute this into Equation 3:
  \[
  z = \left(\frac{1}{2}y - 0.05\right) + 2.10 = \frac{1}{2}y + 2.05
  \]

Now, substitute both $ x $ and $ z $ in terms of $ y $ into Equation 1:
\[
\left(\frac{1}{2}y - 0.05\right) + y + \left(\frac{1}{2}y + 2.05\right) = 82
\]

### Step 3: Solve for $ y $.

Combine like terms:
\[
\frac{1}{2}y + y + \frac{1}{2}y - 0.05 + 2.05 = 82
\]

Simplify:
\[
2y + 2 = 82
\]

Subtract 2 from both sides:
\[
2y = 80
\]

Divide by 2:
\[
y = 40
\]

### Step 4: Find $ x $ and $ z $.

Now that we know $ y = 40 $, substitute this back into Equation 2 and Equation 3:
- For $ x $:
  \[
  x = \frac{1}{2}(40) - 0.05 = 20 - 0.05 = 19.95
  \]
- For $ z $:
  \[
  z = 19.95 + 2.10 = 22.05
  \]

### Final Answer:
- **Your groceries cost:** $19.95
- **Sarah's groceries cost:** $40.00
- **Your other roommate's groceries cost:** $22.05

Would you like further details or have any questions?

---

1. How would the solution change if your groceries were more expensive than half of Sarah's groceries?
2. What would happen to the total if Sarah's groceries were reduced by $10?
3. Can you express the total cost of groceries in terms of one roommate's cost only?
4. How could you check the correctness of these calculations?
5. What if you had a discount on your groceries, how would that affect the system of equations?

**Tip:** When creating systems of equations, clearly define variables to represent unknowns for easier substitution and solving.

Learn how to solve a system of equations to determine individual grocery costs. This problem involves linear equations and systems of equations, suitable for students in Grades 7-9.

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