Problem 33 - College Algebra Essentials

Question

Use a system of linear equations in three variables to solve In this exercise, we refer to annual spending per person in 2010 dollar. The combined spending on housing, vehicles/gas, and health care was 3,840 dollar. The difference between spending on housing and spending on vehicles/gas was 3864 dollar. The difference between spending on housing and spending on health care was 695 dollar. Find the average per-person spending on housing, vehicles/gas, and health care in 2010 . (BAR GRAPH CAN'T COPY)

Step-by-Step Solution

Answer

The solution obtained is \(v = -512\), \(m = 1169\), and \(h = 4183\). Since cost cannot be negative, there might be a mistake in the formulation of the problem or the computations.

Step 1: Formulate the System of Linear Equations

Let's denote housing as \(h\), vehicles/gas as \(v\), and health care as \(m\). From the problem we have the following equations: \[ h + v + m = 3840 \] \[ h - v = 3864 \] \[ h - m = 695 \]

Step 2: Solve the System of Linear Equations

First, solve the second equation for \(h\): \(h = v + 3864\). Substitute this into the first and the third equations: \[ v + 3864 + v + m = 3840 \] (first equation) and \[ v + 3864 - m = 695 \] (third equation). You can solve the first equation to find \(v\), and then the second equation to find \(m\). Finally, substitute \(v\) and \(m\) in the first original equation to find \(h\).

Step 3: Compute the Results

Solving the above gives \(v = -512\), \(m = 1169\), and \(h = 4183\). However, cost cannot be negative, so there might be a mistake in the formulation of the problem as \(v\) turned out negative. Check the formulation of the system and/or the computations, or ask for clarification.

Key Concepts

Linear Equations in Three Variables

A system of linear equations in three variables involves finding values for three unknowns that satisfy several given equations. These equations are formed from relationships or constraints that are expressed mathematically.

• Each equation typically includes three variables, often represented by letters such as \(x\), \(y\), and \(z\), or in this case \(h\), \(v\), and \(m\), representing housing, vehicles/gas, and healthcare spending respectively.
• The goal is to find a set of values that satisfy all the given equations simultaneously. This means substituting the found values back into each equation should hold true, without contradiction.
• When forming equations, it is crucial to correctly interpret the relationships described in word problems to translate them into valid mathematical expressions. Accurate representation is critical to solving for the variables successfully.

Understanding the nature of these equations helps in approaching practical problems where multiple influences or factors need to be considered together. By practicing solving these types of systems, learners can improve their algebraic skills and problem-solving abilities, which are vital in both academic and real-world contexts.

Problem Solving in Algebra

Problem-solving in algebra is a vital skill that involves understanding the problem, formulating equations, and finding solutions. It requires logical thinking and the ability to translate words into mathematical expressions.

• Initial steps include identifying the unknowns, defining them with variables, and establishing relationships between them.
• Once equations are formulated, you need to solve them systematically. Techniques include substitution, elimination, or matrix methods for larger systems. These methods help isolate each variable and solve for their values.
• Once potential solutions are found, it's crucial to verify them. Plugging the solutions back into the original equations ensures they're correct and confirms understanding.
• Interpretation of solutions is also critical. The outcomes should make logical sense within the context of the problem. For instance, realizing that a negative value for spending is unfeasible can point to potential errors in the initial formulation.

Mastering these problem-solving skills not only aids academic success in algebra but also strengthens logical reasoning required in everyday decision-making.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world situations. This enables the analysis and solution of complex problems using mathematical tools.

• The process begins by interpreting the problem's context and accurately representing it using equations that reflect given data and constraints.
• In the presented exercise, expenditures on housing, vehicles/gas, and healthcare are modeled by a system of equations. The relationships given, like total spending and differences between categories, dictate how each equation is structured.
• Once a model is defined, you solve it using suitable mathematical techniques. This might involve linear algebra tools, graphical methods, or computational algorithms.
• Finally, the solution needs interpretation. Results from the model should be able to answer the initial real-world question meaningfully. Discrepancies in results, such as negative expenditures, prompt re-evaluation of the model or the question itself.

Engaging with mathematical modeling not only aids in solving specific problems but also enhances understanding of how mathematics can be applied holistically across various domains.