Questions: Yarita and her children went into a restaurant that sells hamburgers for 5 each and tacos for 2.50 each. Yarita has 60 to spend and must buy at least 14 hamburgers and tacos altogether. If x represents the number of hamburgers purchased and y represents the number of tacos purchased, write and solve a system of inequalities graphically and determine one possible solution.

Yarita and her children went into a restaurant that sells hamburgers for 5 each and tacos for 2.50 each. Yarita has 60 to spend and must buy at least 14 hamburgers and tacos altogether. If x represents the number of hamburgers purchased and y represents the number of tacos purchased, write and solve a system of inequalities graphically and determine one possible solution.

Transcript text: Yaritza and her children went into a restaurant that sells hamburgers for $5 each and tacos for $2.50 each. Yaritza has $60 to spend and must buy at least 14 hamburgers and tacos altogether. If $x$ represents the number of hamburgers purchased and $y$ represents the number of tacos purchased, write and solve a system of inequalities graphically and determine one possible solution.

Solution

Solution Steps

Step 1: Define the Variables

Let $ x $ represent the number of hamburgers purchased and $ y $ represent the number of tacos purchased.

Step 2: Formulate the Inequalities

The total cost constraint: $ 5x + 2.5y \leq 60 $
The total number of items constraint: $ x + y \geq 14 $

Step 3: Graph the Inequalities

Convert the inequalities to equations to find boundary lines:
- $ 5x + 2.5y = 60 $
- $ x + y = 14 $
Plot the boundary lines on the graph:
- For $ 5x + 2.5y = 60 $:
  - When $ x = 0 $, $ y = 24 $
  - When $ y = 0 $, $ x = 12 $
- For $ x + y = 14 $:
  - When $ x = 0 $, $ y = 14 $
  - When $ y = 0 $, $ x = 14 $
Shade the feasible region:
- For $ 5x + 2.5y \leq 60 $, shade below the line.
- For $ x + y \geq 14 $, shade above the line.