Problem 34 - Elementary Algebra Within Reach

Question

A person plans to invest up to \(\$ 10,000\) in two different interest-bearing accounts, account \(X\) and account \(Y\). Account \(Y\) is to contain at most \(\$ 3000\). Moreover, account \(X\) should have at least three times the amount in account Y. Write a system of linear inequalities that describes the various amounts that can be deposited in each account, and sketch the graph of the system.

Step-by-Step Solution

Answer

The inequalities that represent the possible amounts to be deposited in each account are as follows: \(x + y \leq 10000\), \(y \leq 3000\), and \(x \geq 3y\). These inequalities can be plotted on a graph, and the intersection area of these inequalities represents the possible amounts that could be deposited in each account.

Step 1: Define Variables

It's helpful to start by defining the variables that are going to be used. In this case, let \(x\) represent the amount invested in account \(X\), and \(y\) represent the amount invested in account \(Y\).

Step 2: Converting Conditions Into Inequalities

The conditions stated in the problem can be formulated as linear inequalities. Firstly, 'a person plans to invest up to \(\$ 10,000\)' in both accounts, which can be expressed as \(x + y \leq 10000\). Secondly, 'account \(Y\) is to contain at most \(\$ 3000\) can be expressed as \(y \leq 3000\). Thirdly, 'account \(X\) should have at least three times the amount in account \(Y\)' translates to \(x \geq 3y\).

Step 3: Sketch the Graph

Sketching inequalities on a graph involves using the inequality to find a boundary line, and then shading in the area that satisfies the inequality. To graph these inequalities, first plot the line \(x + y = 10000\). This is the boundary for the first inequality, so shade in the area that includes values of \(x\) and \(y\) such that \(x + y\) is less than or equal to \(10000\). Repeat the process for the other two inequalities, \(y = 3000\) and \(x = 3y\). The intersection area of all three inequalities represents the possible amounts that could be invested in each account.

Key Concepts

Investments

An investment is when you allocate your money with the expectation of generating a financial return. In the context of the problem, this involves placing money in two accounts: account X and account Y. The goal is to manage your investments in a way that adheres to particular conditions, such as not exceeding a total of $10,000 across both accounts. This strategy requires careful planning to ensure the best balance between these accounts to meet various financial goals and constraints.

Variables

Variables are symbols used to represent unknown values that can change within the context of a problem. In this exercise, we utilize variables to handle the money invested in different accounts.

• Let \( x \) be the amount invested in account X.
• Let \( y \) be the amount invested in account Y.

These variables help us translate real-world constraints into mathematical statements so that we can clearly compute and visualize feasible solutions.

Graphing Inequalities

Graphing inequalities involves plotting boundary lines on a graph and determining the area that satisfies each inequality. For the given problem, the graph helps us visualize the range of possible investments:

• First, plot \( x + y = 10000 \) which represents the maximum combined investment.
• Next, plot the line \( y = 3000 \), limiting the amount in account Y.
• Lastly, \( x = 3y \) is plotted to ensure that account X remains at least three times account Y's amount.

The intersection of shaded areas from these boundaries shows the permissible funds distribution for accounts X and Y.

Systems of Inequalities

A system of inequalities consists of multiple inequalities that impose various limits or conditions on variables. Each inequality represents a rule that the solution must satisfy. In our investment scenario, there are three conditions:

• Total investments must be less than or equal to \( \\(10,000 \): \( x + y \leq 10000 \).
• Investment in account Y must not exceed \( \\)3,000 \): \( y \leq 3000 \).
• Investment in account X must be at least three times that of account Y: \( x \geq 3y \).

The solution to this system is the set of all \( (x, y) \) pairs that satisfy all three inequalities, depicted as the overlapping region on the graph.