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Chapter 4: Problem 12

A restaurant purchased eight tablecloths and five napkins for \(\$ 106 .\) A week later, a tablecloth and six napkins were bought for \(\$ 24 .\) Find the cost of one tablecloth and the cost of one napkin, assuming the same prices for both purchases.

Short Answer

Expert verified
The cost of one tablecloth is $12 and the cost of one napkin is $2.

Step by step solution

01

Set up the equations

Let's denote T as the cost of one tablecloth and N as the cost of one napkin. According to the problem, we have two equations: \[8T + 5N = 106\] (as eight tablecloths and five napkins cost $106), and \[T + 6N = 24\] (as a tablecloth and six napkins cost $24).
02

Solve the system of equations

To solve the system, it's logical to express one of the variables from the second equation, as it contains smaller coefficients. Let's express T: \[T = 24 - 6N,\] then substitute T into the first equation: \[8(24-6N) + 5N = 106.\] After simplifying, you get \[192 - 48N + 5N = 106. \] Collecting like terms yields \[-43N = -86.\] Hence, dividing both sides by -43, we find that N = 2.
03

Find the other variable

Now we substitute N = 2 into the second equation (T = 24 - 6*2) to find the value for T. This gives us that T = 12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are fundamental in algebra and describe relationships with constant rates of change. Each equation involves variables raised only to the first power. In our exercise, we had two equations like this:
  • 8T + 5N = 106
  • T + 6N = 24
Both equations express costs in terms of tablecloths (T) and napkins (N). The goal is to find a single unique solution, meaning the exact cost of one tablecloth and one napkin. Linear equations form straight lines when graphed, and a solution to the system is where they intersect.
System of Equations
A system of equations is a collection of two or more equations with the same set of variables. Solving a system of equations means finding a set of variable values that make all equations true simultaneously. In our problem, we dealt with a system composed of two linear equations:
  • Equation A: 8T + 5N = 106
  • Equation B: T + 6N = 24
We solve them together to determine the costs of one tablecloth and one napkin. Various methods, such as substitution or elimination, can be applied to find the solution. The essence is to work with both equations at once to reveal the values of the variables.
Variable Substitution
Variable substitution is an essential technique used to solve systems of equations. It involves replacing one variable with an expression containing the other variables. In our task, we used substitution by solving for T in the second equation:
  • T = 24 - 6N
Then, we replaced T in the first equation with this expression:
  • 8(24-6N) + 5N = 106
By doing this, we transformed the first equation into one that only involved N, making it straightforward to solve. Substitution simplifies complex equations into more manageable ones, helping us isolate each variable individually.
Problem Solving Steps
Solving algebraic problems requires a strategic approach. Here are the steps we followed in this exercise:
  • Identify the variables: Understand what each symbol represents in the context, such as T for the cost of a tablecloth and N for a napkin.
  • Set up equations: Translate the problem's words into mathematical equations based on given conditions.
  • Solve the system: Use methods like substitution or elimination to find the variable values.
  • Verify the solution: Substitute the values back into the original equations to ensure they satisfy all given conditions.
This structured approach helps in efficiently tackling algebra problems, whether simple or complex. It's crucial to verify solutions to confirm they are indeed correct and satisfy the initial problem conditions.

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Most popular questions from this chapter

Things did not go quite as planned. You invested \(\$ 20,000\), part of it in a stock with a \(12 \%\) annual return. However, the rest of the money suffered a \(5 \%\) loss. If the total annual income from both investments was \(\$ 1890,\) how much was invested at each rate?

Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. A person invested \(\$ 6700\) for one year, part at \(8 \%,\) part at \(10 \%,\) and the remainder at \(12 \% .\) The total annual income from these investments was \(\$ 716 .\) The amount of money invested at \(12 \%\) was \(\$ 300\) more than the amounts invested at \(8 \%\) and \(10 \%\) combined. Find the amount invested at each rate.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I think of equilibrium as the point at which quantity demanded exceeds quantity supplied.

The manager of a candystand at a large multiplex cinema has a popular candy that sells for \(\$ 1.60\) per pound. The manager notices a different candy worth \(\$ 2.10\) per pound that is not selling well. The manager decides to form a mixture of both types of candy to help clear the inventory of the more expensive type. How many pounds of each kind of candy should be used to create a 75 -pound mixture selling for \(\$ 1.90\) per pound?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Unlike substitution, the addition method lets me see solutions as intersection points of graphs.
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