Question

Restaurant Management A restaurant manager wants to purchase 200 sets of dishes. One design costs \(\$ 25\) per set, and another costs \(\$ 45\) per set. If she has only \(\$ 7400\) to spend, how many sets of each design should she order?

Short answer

80 sets of the first design and 120 sets of the second design.

Step-by-step solution

- Define Variables

Let x be the number of sets of the first design (costs \(25) and y be the number of sets of the second design (costs \)45).

- Set up the Equations

We have two conditions:1. The total number of sets is 200: \( x + y = 200 \)2. The total cost is $7400: \( 25x + 45y = 7400 \)

- Solve the System of Equations

Solve the first equation for one variable:\( y = 200 - x \)Substitute into the second equation:\( 25x + 45(200 - x) = 7400 \)

- Simplify and Solve for x

Simplify the equation:\( 25x + 9000 - 45x = 7400 \) Combine like terms:\( -20x + 9000 = 7400 \) Isolate x:\( -20x = -1600 \) \( x = 80 \)

- Solve for y

Substitute back into the first equation:\( y = 200 - x \)\( y = 200 - 80 \)\( y = 120 \)

- Verify the Solution

Check the solution in the cost equation:\( 25(80) + 45(120) = 2000 + 5400 = 7400 \)The solution is verified.

Key concepts

linear equations

Linear equations are fundamental tools in mathematics. They help us model relationships between different quantities using straight-line graphs. A linear equation typically has the format:
\[ ax + by = c \]
where:

• \(x\) and \(y\) are the variables.
• \(a\), \(b\), and \(c\) are constants.

In a system of linear equations, we use more than one linear equation together to find solutions that satisfy all equations involved. Using the problem about the restaurant manager, we can see that we are dealing with such a system:

• \( x + y = 200 \) (representing the total number of sets)
• \( 25x + 45y = 7400 \) (representing the total cost)

By solving these equations together, we can find specific values for \(x\) and \(y\) (the number of each type of dish set). Learning how to handle linear equations is essential for tackling real-world problems, especially in fields like finance and resource management.

substitution method

The substitution method is a common technique used to solve systems of linear equations. The main idea is to solve one of the equations for one variable and then substitute that expression into the other equation. Here’s the step-by-step guide:

• Solve for a Variable: In the restaurant problem, we first solved the equation \( x + y = 200 \) for \(y\), finding \( y = 200 - x \).


• Substitute: Then, we substituted that into the second equation: \(25x + 45(200 - x) = 7400 \).


• Simplify: Simplify the equation to isolate the variable, \(x\). This allows us to solve for \( x\): \( 25x + 9000 - 45x = 7400 \)


• Solve: Combine like terms and solve for \( x \), giving us \( x = 80 \).


• Back-Substitute: Finally, substitute the value of \( x \) back into the first equation to find the value of \( y \):\( y = 200 - 80 = 120 \).

The substitution method helps break the problem into manageable steps, making it easier to find the solution.

real-world applications

Understanding how to solve linear systems using methods like substitution is very useful in real-world scenarios. For instance, the restaurant manager's problem is a common example where these skills come into play. Here are more applications:

• Business Planning: Companies often need to allocate resources wisely. Planning production levels and costs, just like our restaurant manager, requires solving linear equations.


• Finance: Budgeting and financial forecasting involve solving systems of equations to determine how to distribute funds or predict profits.


• Engineering: Engineers use linear systems to design and optimize systems and structures, ensuring they meet all necessary criteria.


• Everyday Decisions: Even in daily life, whether planning your expenses or deciding the mixture of groceries to buy under a budget, you are using linear equations unconsciously.

By mastering linear equations and methods like substitution, you are better equipped to handle numerous practical and professional challenges.