Josiah and his children went into a bakery that sells cupcakes for $4 each and brownies for $2.50 each. Josiah has $35 to spend and must buy a minimum of 11 cupcakes and brownies altogether. If xx represents the number of cupcakes purchased and yy represents the number of brownies purchased, write and solve a system of inequalities graphically and determine one possible solution.

We are asked to write and solve a system of inequalities based on the given problem, where Josiah has $35 to spend on cupcakes and brownies, and must buy at least 11 items in total. The prices and constraints will help us form the inequalities.

### Step 1: Define the variables
Let:
- $ x $ be the number of cupcakes purchased.
- $ y $ be the number of brownies purchased.

### Step 2: Write the system of inequalities

#### 1. Budget constraint:
Each cupcake costs $4 and each brownie costs $2.50. Josiah has $35 to spend, so the total cost of the cupcakes and brownies must be less than or equal to $35:
\[
4x + 2.5y \leq 35
\]

#### 2. Minimum number of items constraint:
Josiah must buy at least 11 items (cupcakes and brownies combined), so:
\[
x + y \geq 11
\]

#### 3. Non-negative quantities:
Josiah can't buy negative quantities of cupcakes or brownies, so:
\[
x \geq 0 \quad \text{and} \quad y \geq 0
\]

### Step 3: Solve the system graphically

We now graph the system of inequalities:
1. For the budget constraint, $ 4x + 2.5y \leq 35 $:
   - Solve for $ y $ in terms of $ x $: 
     \[
     y \leq \frac{35 - 4x}{2.5}
     \]
   - Find intercepts:
     - When $ x = 0 $, $ y = 14 $ (since $ 2.5y = 35 $).
     - When $ y = 0 $, $ x = 8.75 $ (since $ 4x = 35 $, or approximately 9 for a practical answer).

2. For the minimum number of items constraint, $ x + y \geq 11 $:
   - Solve for $ y $: 
     \[
     y \geq 11 - x
     \]
   - This is a line with slope -1, passing through points $ (0, 11) $ and $ (11, 0) $.

### Step 4: Graphical Solution
To solve graphically, plot the two inequalities and find the region where the solutions overlap, taking into account $ x \geq 0 $ and $ y \geq 0 $.

### Step 5: Determine one possible solution
The overlapping region will give all possible values of $ x $ and $ y $. One possible solution from this system could be:
\[
x = 6 \quad \text{(cupcakes)}, \quad y = 5 \quad \text{(brownies)}.
\]
This solution satisfies both:
- $ 6 + 5 = 11 $ (at least 11 items), and
- $ 4(6) + 2.5(5) = 24 + 12.5 = 36.5 $, which slightly exceeds $35, so it's not a valid solution.

Trying another example might help! For instance

Josiah and his children went into a bakery that sells cupcakes for $4 each and brownies for $2.50 each. Josiah has $35 to spend and must buy a minimum of 11 cupcakes and brownies altogether. If x represents the number of cupcakes purchased and y represents the number of brownies purchased, write and solve a system of inequalities graphically and determine one possible solution.

Learn how to solve a system of inequalities involving purchasing cupcakes and brownies with a budget of $35. This problem covers key algebraic concepts, including linear inequalities and graphing solutions to find feasible purchases. Suitable for students in Grades 8-10.

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