Step 1: Identify Income and Expenses

Income Sources

1. Allowance: Suppose your weekly allowance is $50.
2. Part-time Job Earnings: Assume you earn $80 per week from a part-time job.
3. Other Sources: Consider any other sources of income, such as occasional babysitting, amounting to $20 per week.

Total Weekly Income (I):
$I = 50 + 80 + 20 = 150 \text{ dollars}$

Expense Categories

1. Food: $30 per week
2. Transportation: $20 per week
3. Entertainment: $25 per week
4. Savings: $50 per week
5. Other Expenses: $15 per week

Total Weekly Expenses (E):
$E = 30 + 20 + 25 + 50 + 15 = 140 \text{ dollars}$

Step 2: Create a Linear Equation

Let $I$ be the total income and $E$ be the total expenses. The budgeting equation ensuring you stay within your financial limits is:

$I = E$

Given the values:

$150 = 140$

This shows you have a surplus of $10, which can be allocated to savings or other needs.

Step 3: Graph the Budget Function

1. Graph Setup:

   • Place Income (I) on the y-axis.
   • Place Expenses (E) on the x-axis.

2. Plotting Points:

   • Plot the point $(140, 150)$ since your expenses are $140, and income is $150.
   • Draw the line $I = E$, which is the 45-degree line from the origin, representing where income equals expenses.

3. Feasible Region:

   • Shade the region above the line $I = E$ where $I \geq E$. This region represents scenarios where you have a surplus.

Step 4: Analyze Changes in Income or Expenses

If your allowance increases by $10:

• New Total Income:
  $I = 160$ dollars

• New Linear Equation:
  $I = 160$

• Graph and Analysis:
  Plot the new point $(140, 160)$ and observe that the surplus has increased, allowing more flexibility in your budget.

Step 5: Prepare the Budget Plan and Presentation

1. Purpose:
   The goal is to create a balanced budget to manage income and expenses efficiently.

2. Detailed List of Income and Expenses:
   Clearly list your sources of income and categorize your expenses.

3. Graph:
   Include the graph showing the relationship between income and expenses, highlighting the feasible region.

4. Impact of Changes:
   Discuss how an increase in income or unexpected expenses might affect your budget.

5. Summary:
   A balanced budget was achieved with a $10 surplus, allowing for savings or additional spending.

6. Graphs and Charts:
   Include all relevant graphs and calculations to support your plan.

Questions for Further Exploration:

1. How would an unexpected expense affect your budget?
2. What strategies can you use if your income decreases?
3. How can you prioritize savings in your budget?
4. What are the benefits of tracking your expenses regularly?
5. How can you adjust your budget if your income becomes irregular?

Tip: Always allocate a portion of your income to savings to prepare for unexpected expenses.