Problem 78 - College Algebra

Question

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Based on the graph. (GRAPH CAN'T COPY) A new restaurant is to contain two-seat tables and four-seat tables. Fire codes limit the restaurant's maximum occupancy to 56 customers. If the owners have hired enough servers to handle 17 tables of customers, how many of each kind of table should they purchase?

Step-by-Step Solution

Answer

The restaurant should have 6 two-seat tables and 11 four-seat tables.

Step 1: Formulate the Equations

We formulate the two equations using the conditions given: 1. Each 2-seat table (x) and 4-seat table (y) combined should be able to accommodate maximum of 56 customers. This leads to our first equation, \(2x + 4y = 56\).2. The total number of tables, that is, the number of 2-seat tables (x) and 4-seat tables (y), should be 17. This gives us our second equation \(x + y = 17\).

Step 2: Solve Equation 2 for one Variable

From the second equation \(x + y = 17\), isolate y: \(y = 17 - x\).

Step 3: Substitute

Now, substitute \(y = 17 - x\) into the first equation \(2x + 4y = 56\). This gives \(2x + 4(17 - x) = 56\).

Step 4: Simplification and Solving for x

Simplifying the equation gives \(2x + 68 - 4x = 56\). After combining like terms, we get \(-2x = -12\). Solving this for x gives \(x = 6\).

Step 5: Solving for y

Substitute \(x = 6\) back into the equation \(y = 17 - x\). This gives \(y = 17 - 6 = 11\).

Step 6: Interpret the Solution

So the owners should purchase 6 two-seat tables and 11 four-seat tables.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that involve variables with no exponents or powers greater than one. They form straight-line graphs when plotted on a coordinate plane. In this exercise, we deal with a system of linear equations to solve the problem of arranging tables in a restaurant. The primary feature of linear equations is their simplicity due to their structure, typically written in the form of \( ax + by = c \). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. In our example, we derived two linear equations from the given problem:

• The seating constraint: \( 2x + 4y = 56 \)
• The table constraint: \( x + y = 17 \)

These equations allow us to represent complex relationships between variables in a manageable form, making them a powerful tool in problem solving.

Problem Solving

Problem solving using linear equations involves identifying relationships and constraints within the given scenario and expressing them mathematically. For the restaurant tables arrangement problem, the key is understanding the constraints:

• The maximum occupancy of 56 customers.
• The total number of tables, which is 17.

By converting these constraints into a set of linear equations, we use a structured approach to find how many 2-seat and 4-seat tables the restaurant should purchase. The method involves creating equations that reflect the problem's conditions, manipulating these equations to isolate variables, and then solving them step by step to find a viable solution. This structured approach makes complex problems more accessible and easier to solve.
Once you have your equations, substituting values and simplifying help in pinpointing the exact solution.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It provides a way to express and solve equations that involve unknown quantities. In this problem, algebra helps us manipulate the equations to discover the values of the unknown variables, \(x\) and \(y\).
Algebraic manipulation, such as isolating variables and substitution, is crucial in solving systems of equations. For example, we isolate \(y\) from our table equation \( x + y = 17 \), resulting in \( y = 17 - x \). This expresses \(y\) in terms of \(x\), which is then substituted into the seating equation \( 2x + 4y = 56 \).

• This gives us a single equation in one variable, simplifying the problem significantly.
• Solving this gives us \( x = 6 \), which represents the number of 2-seat tables.
• Substituting back to find \(y\) shows \( y = 11 \), representing the number of 4-seat tables.

These techniques are part of the fundamental toolkit of algebra.

Tables Arrangement

Arranging tables in a restaurant involves optimizing available space to meet constraints like occupancy limits and serving capacity. This exercise uses linear equations to determine the best combination of 2-seat and 4-seat tables, ensuring the restaurant adheres to fire safety codes (max occupancy of 56) and available service capacity (17 tables total).

• By structuring the problem with equations, we efficiently calculate the ideal number of each type of table.
• The solution \( x = 6 \) and \( y = 11 \) tells the owners to use 6 tables for smaller parties and 11 for larger ones.

Implementing mathematical problem solving into practical scenarios, such as table arrangement, demonstrates how math is not just theoretical but also an essential tool in efficient and effective planning.
It ensures the restaurant can serve the maximum number of customers, maintaining a balance between available tables and customer comfort.