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Sean has an annual shoe budget of $640. On average, a pair of Jordans costs $160, and a pair of Adidas costs $40.


  1. Create an equation that represents the combinations of shoe purchases that allow Sean to stay in his budget.
  2. Graph the equation for his budget.
  3. Provide 2 combinations that use his entire budget, 1 combination that would be under budget, and 1 combination that would be over his budget.

Verified Answer :

  • Define the budget equation as \(160x + 40y = 640\), which simplifies to \(4x + y = 16\).
  • Determine the graph intercepts at \((4, 0)\) and \((0, 16)\).
  • Identify combinations using the budget exactly: \((2, 8)\) and \((1, 12)\).
  • Identify combinations under budget: \((1, 2)\), and over budget: \((5, 2)\).
  • The final answer for combinations are: exactly \(\boxed{(2, 8), (1, 12)}\); under \(\boxed{(1, 2)}\); over \(\boxed{(5, 2)}\).

Explanation

  1. Budget Equation Analysis

To determine the combinations of shoe purchases for Sean, we start by defining the variables and the budget equation. Let \(x\) represent the number of pairs of Jordans and \(y\) represent the number of pairs of Adidas. The cost of one pair of Jordans is \(\$160\), and the cost of one pair of Adidas is \(\$40\). The total annual shoe budget is \(\$640\). The relationship between these costs and the total budget is given by the linear equation:
\[160x + 40y = 640\]
To simplify this equation, we can divide both sides by \(40\):
\[4x + y = 16\]
This equation describes all combinations \((x, y)\) where Sean uses exactly \(\$640\). Since \(x\) and \(y\) must be non-negative integers (as one cannot buy a negative amount of shoes), we can easily explore different options within his budget.

  1. Graphing the Budget Equation

To graph the equation \(4x + y = 16\), we identify the intercepts. The x-intercept occurs when \(y = 0\):
\[4x + 0 = 16 \implies x = 4\]
The coordinate point is \((4, 0)\).
The y-intercept occurs when \(x = 0\):
\[4(0) + y = 16 \implies y = 16\]
The coordinate point is \((0, 16)\).
By plotting these two points \((4, 0)\) and \((0, 16)\) on a Cartesian plane and drawing a straight line through them, we represent the set of all possible combinations that exactly equal the \(\$640\) budget for \(x \ge 0\) and \(y \ge 0\). Values under the line represent combinations under budget, and values above the line represent combinations over budget.

  1. Identifying Shoe Purchase Combinations

We look for combinations that satisfy the budget constraints:

  • To use the entire budget (exactly \(\$640\)), we solve \(4x + y = 16\):
  • If \(x = 2\), then \(4(2) + y = 16 \implies y = 8\). Combination: \((2, 8)\).
  • If \(x = 1\), then \(4(1) + y = 16 \implies y = 12\). Combination: \((1, 12)\).
  • To be under budget (less than \(\$640\)), we look for a combination where \(4x + y < 16\):
  • Let \(x = 1\) and \(y = 2\). Total cost: \(160(1) + 40(2) = 160 + 80 = \$240\), which is less than \(\$640\). Combination: \((1, 2)\).
  • To be over budget (more than \(\$640\)), we look for a combination where \(4x + y > 16\):
  • Let \(x = 5\) and \(y = 2\). Total cost: \(160(5) + 40(2) = 800 + 80 = \$880\), which is more than \(\$640\). Combination: \((5, 2)\).

Examples

Budgeting is an essential skill for managing personal finances. For example, if you have a set monthly allowance for recreational activities like movies and arcade games, you can create a similar linear equation to find various combinations of activities you can afford. By graphing this relationship, you can visually determine which combinations fit your budget, which are too expensive, and how much "spending room" you have for more items. This helps you make informed choices while ensuring you do not overspend.

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