Gabriella and her children went into a bakery that sells cupcakes for $3 each and donuts for $1.50 each. Gabriella has $30 to spend and must buy at least 13 cupcakes and donuts altogether. If xx represents the number of cupcakes purchased and yy represents the number of donuts purchased, write and solve a system of inequalities graphically and determine one possible solution.

Let's define the variables:
- $ x $ represents the number of cupcakes purchased.
- $ y $ represents the number of donuts purchased.

### Step 1: Setting Up the Inequalities
1. **Total cost constraint**: Each cupcake costs $3, and each donut costs $1.50. Gabriella has $30 to spend, so the total cost must not exceed $30:
   \[
   3x + 1.5y \leq 30
   \]

2. **Quantity constraint**: Gabriella must buy at least 13 cupcakes and donuts combined:
   \[
   x + y \geq 13
   \]

3. **Non-negativity constraint**: The number of cupcakes and donuts cannot be negative:
   \[
   x \geq 0, \quad y \geq 0
   \]

### Step 2: Solving the System Graphically
To solve the system graphically, we will plot the inequalities:

1. **Graphing $ 3x + 1.5y \leq 30 $**:
   - Rewrite the inequality as $ y \leq \frac{30 - 3x}{1.5} $.
   - Find the intercepts:
     - When $ x = 0 $, $ y = \frac{30}{1.5} = 20 $.
     - When $ y = 0 $, $ x = \frac{30}{3} = 10 $.
   - Plot these points $(0, 20)$ and $(10, 0)$, and shade below the line.

2. **Graphing $ x + y \geq 13 $**:
   - Rewrite the inequality as $ y \geq 13 - x $.
   - Find the intercepts:
     - When $ x = 0 $, $ y = 13 $.
     - When $ y = 0 $, $ x = 13 $.
   - Plot these points $(0, 13)$ and $(13, 0)$, and shade above the line.

3. **Graphing $ x \geq 0 $ and $ y \geq 0 $**:
   - These inequalities indicate that the solution must be in the first quadrant.

### Step 3: Determining One Possible Solution
Find a point in the region where all shaded areas overlap. One such point is:
   \[
   (x, y) = (8, 7)
   \]
This solution represents buying 8 cupcakes and 7 donuts.

### Verification
1. **Total cost**: $ 3(8) + 1.5(7) = 24 + 10.5 = 34.5 $, which does not satisfy the total budget constraint.
So, we need another point within the solution range.

Gabriella and her children went into a bakery that sells cupcakes for $3 each and donuts for $1.50 each. Gabriella has $30 to spend and must buy at least 13 cupcakes and donuts altogether. If x represents the number of cupcakes purchased and y represents the number of donuts purchased, write and solve a system of inequalities graphically and determine one possible solution.

Learn how to solve a system of inequalities involving a budget of $30 for buying cupcakes and donuts, using graphical methods. This problem explores algebraic concepts and is suitable for students in Grades 8-10. Includes step-by-step plotting and analysis.

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