Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. Today's cafeteria specials at a high school in Stafford are a deluxe t... Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. Today's cafeteria specials at a high school in Stafford are a deluxe turkey sandwich and a chef salad. During early lunch, the cafeteria sold 15 turkey sandwiches and 45 chef salads, for a total of $165. During the late lunch, 41 turkey sandwiches and 59 chef salads were sold, for a total of $259. How much does each item cost? A turkey sandwich costs $___, and a chef salad costs $___.
Understand the Problem
The question is asking to set up a system of equations based on the sales of turkey sandwiches and chef salads in two lunch periods and then solve for their costs using the elimination method.
Answer
A turkey sandwich costs $2, and a chef salad costs $3.
Answer for screen readers
A turkey sandwich costs $2, and a chef salad costs $3.
Steps to Solve
-
Define Variables
Let $x$ represent the cost of a turkey sandwich and $y$ represent the cost of a chef salad. -
Set Up the Equations
From the early lunch data:
15 turkey sandwiches and 45 chef salads total $165.
This gives us the equation:
$$ 15x + 45y = 165 $$
From the late lunch data:
41 turkey sandwiches and 59 chef salads total $259.
This gives us the equation:
$$ 41x + 59y = 259 $$
-
Simplify the First Equation
We can simplify the first equation by dividing everything by 15:
$$ x + 3y = 11 $$
(Educational insight: This makes calculations easier later on.) -
Write the Simplified System of Equations
We have now:
- $$ x + 3y = 11 $$
- $$ 41x + 59y = 259 $$
-
Elimination Method
To eliminate $x$, multiply the first equation by 41:
$$ 41(x + 3y) = 41(11) $$
This gives:
$$ 41x + 123y = 451 $$ -
Set Up the New System
Now we have:
- $$ 41x + 123y = 451 $$
- $$ 41x + 59y = 259 $$
-
Subtract the Second Equation from the First
Subtract the second equation from the first to eliminate $x$:
$$ (41x + 123y) - (41x + 59y) = 451 - 259 $$
This simplifies to:
$$ 64y = 192 $$ -
Solve for $y$
Dividing both sides by 64 gives:
$$ y = 3 $$ -
Substitute Back to Find $x$
Now substitute $y = 3$ into one of the original equations, using the first:
$$ x + 3(3) = 11 $$
Thus:
$$ x + 9 = 11 $$
Therefore:
$$ x = 2 $$
A turkey sandwich costs $2, and a chef salad costs $3.
More Information
These equations illustrate a real-world application of systems of equations, specifically in a cafeteria setting to calculate costs based on sales data. This method ensures accurate pricing for budget management.
Tips
- Not simplifying equations: Failing to simplify can lead to complex calculations that are prone to errors. Always look for common factors.
- Sign errors: Be careful with signs when performing operations, especially when adding or subtracting equations.
- Neglecting to check: Forgetting to verify your solutions by substituting back into the original equations can lead to incorrect assumptions.
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