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Chapter 2: Problem 56

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. Cantwell Associates, a real estate developer, is planning to build a new apartment complex consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 192 units is planned, and the number of family units (two- and three-bedroom townhouses) will equal the number of one-bedroom units. If the number of one-bedroom units will be 3 times the number of threebedroom units, find how many units of each type will be in the complex.

Short Answer

Expert verified
In the apartment complex, there will be 96 one-bedroom units, 64 two-bedroom townhouses, and 32 three-bedroom townhouses.

Step by step solution

01

Define the variables

Let x represent the number of one-bedroom units, y represent the number of two-bedroom townhouses, and z represent the number of three-bedroom townhouses.
02

Set up the equations

We can set up the following system of linear equations: 1. Total number of units: \(x + y + z = 192\) 2. Number of family units equal to the number of one-bedroom units: \(y + z = x\) 3. Number of one-bedroom units is 3 times the number of three-bedroom units: \(x = 3z\)
03

Substitute equations to solve for variables

We can substitute equation (3) into equation (2) to solve for y and z: \(y + z = 3z\) or \(y = 2z\) Now we can substitute equation (3) and the new expression for y into equation (1): \(3z + 2z + z = 192\)
04

Solve for z

Combining the terms in the equation, we get: \(6z = 192\) Divide both sides by 6: \(z = 32\) Now that we have the value for z, we can find the values for x and y.
05

Solve for x and y

Using equation (3), we can find the value of x: \(x = 3z = 3(32) = 96\) Now, using the expression for y, we can find the value of y: \(y = 2z = 2(32) = 64\)
06

State the solution

In the apartment complex, there will be 96 one-bedroom units, 64 two-bedroom townhouses, and 32 three-bedroom townhouses.

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Key Concepts

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

System of Equations
A system of equations is a collection of equations with a shared set of unknowns. When it comes to linear equations in three variables, like in this exercise, the objective is to find values for each variable that satisfy all the equations simultaneously. These types of problems are common in algebra and real-world applications, such as planning construction projects.

In our example, three different types of apartment units need to be calculated using a system of three equations. These equations correspond to distinct relationships established in the problem description. Here are the equations again:
  • The total number of units, which is a combination of all types: \(x + y + z = 192\).
  • The number of one-bedroom units equals the number of family units (two- and three-bedrooms combined): \(y + z = x\).
  • Three-bedroom units are one-third of the one-bedroom units: \(x = 3z\).
To solve a system of equations efficiently, you usually start by expressing one variable in terms of another and then substitute into the remaining equations. By following a step-by-step approach, you can find the solution that satisfies all given conditions.
Real Estate Problem Solving
Real estate development often involves various types of problem-solving, especially when planning the composition of new developments. Solving these types of problems can help developers make decisions about how many units of each type to construct. This ensures that their projects are viable, meet market demands, and comply with strategic goals.

In this specific problem, we're determining the mix of different unit types in a new apartment complex. Each type of unit has its influence on potential income, costs, and appeal to different market segments. By setting up equations based on logical constraints, such as the total number of units and relationships between unit types, developers can efficiently plan their projects.

This type of problem-solving is crucial for ensuring that projects are not only feasible but also profitable and appealing to target audiences. It requires analytical thinking and an understanding of market demands, thus making real estate both a science and an art.
Step-by-Step Solution
A step-by-step solution methodically breaks down complex problems into manageable parts, which is very helpful in understanding how to navigate systems of equations. Following these steps provides clarity and aids in grasping the overall problem-solving strategy.

In the exercise solution:
  • First, the variables are clearly defined, indicating the need to establish a foundation.
  • Next, equations are generated based on logical interpretations of the problem.
  • Substitution is used to simplify equations, making them easier to solve sequentially.
  • Once a variable is solved, others are found using expressions already established, leading to the solution.
This approach ensures that each component of the problem is tackled methodically, ensuring a better understanding and eliminating confusion. Providing such detailed solutions promotes deeper learning and builds confidence in students' problem-solving abilities.

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Most popular questions from this chapter

Bob, a nutritionist who works for the University Medical Center, has been asked to prepare special diets for two patients, Susan and Tom. Bob has decided that Susan's meals should contain at least \(400 \mathrm{mg}\) of calcium, \(20 \mathrm{mg}\) of iron, and \(50 \mathrm{mg}\) of vitamin \(\mathrm{C}\). whereas Tom's meals should contain at least \(350 \mathrm{mg}\) of calcium, \(15 \mathrm{mg}\) of iron, and \(40 \mathrm{mg}\) of vitamin \(\mathrm{C}\). Bob has also decided that the meals are to be prepared from three basic foods: food \(\mathrm{A}\), food \(\mathrm{B}\), and food \(\mathrm{C}\). The special nutritional contents of these foods are summarized in the accompanying table. Find how many ounces of each type of food should be used in a meal so that the minimum requirements of calcium, iron, and vitamin \(\mathrm{C}\) are met for each patient's meals. $$ \begin{array}{lccc} \hline && {\text { Contents (mg/oz) }} & \\ & \text { Calcium } & \text { Iron } & \text { Vitamin C } \\ \hline \text { Food A } & 30 & 1 & 2 \\ \hline \text { Food B } & 25 & 1 & 5 \\ \hline \text { Food C } & 20 & 2 & 4 \\ \hline \end{array} $$

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{rr} x_{1}-x_{2}+3 x_{3}= & 14 \\ x_{1}+x_{2}+x_{3}= & 6 \\ -2 x_{1}-x_{2}+x_{3}= & -4 \end{array} $$

A university admissions committee anticipates an enrollment of 8000 students in its freshman class next year. To satisfy admission quotas, incoming students have been categorized according to their sex and place of residence. The number of students in each category is given by the matrix $$ \begin{array}{l} \text { In-state } \\ \text { A= Out-of-state } \\ \text { Foreign } \end{array}\left[\begin{array}{rr} 2700 & 3000 \\ 800 & 700 \\ 500 & 300 \end{array}\right] $$ By using data accumulated in previous years, the admissions committee has determined that these students will elect to enter the College of Letters and Science, the College of Fine Arts, the School of Business Administration, and the School of Engineering according to the percentages that appear in the following matrix: $$ B=\begin{array}{l} \text { Male } \\ \text { Female } \end{array}\left[\begin{array}{llll} 0.25 & 0.20 & 0.30 & 0.25 \\ 0.30 & 0.35 & 0.25 & 0.10 \end{array}\right] $$ Find the matrix \(A B\) that shows the number of in-state, outof-state, and foreign students expected to enter each discipline.

K & R Builders build three models of houses, \(M_{1}, M_{2}\), and \(M_{3}\), in three subdivisions I, II, and III located in three different areas of a city. The prices of the houses (in thousands of dollars) are given in matrix \(A\) : K\& R Builders has decided to raise the price of each house by \(3 \%\) next year. Write a matrix \(B\) giving the new prices of the houses.

Matrix \(A\) is an input-output matrix associated with an economy, and matrix \(D\) (units in millions of dollars) is a demand vector. In each problem,find the final outputs of each industry such that the demands of industry and the consumer sector are met. $$ A=\left[\begin{array}{ll} 0.4 & 0.2 \\ 0.3 & 0.5 \end{array}\right] \text { and } D=\left[\begin{array}{l} 12 \\ 24 \end{array}\right] $$
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