Problem 34 - College Algebra

Question

Use a system of linear equations in three variables to solve Exercises \(33-4 I\) The bar graph shows the average annual spending per person on selected items in 1980 and \(2010 .\) All dollar amounts are adjusted for inflation. Use this display to solve Exercises \(33-34\) In this exercise, we refer to annual spending per person in \(1980 .\) The combined spending on housing, vehicles/gas, and health care was \(\$ 7073 .\) The difference between spending on housing and spending on vehicles/gas was \(\$ 1247 .\) The difference between spending on housing and spending on health care was \(\$ 1466 .\) Find the average per-person spending on housing. vehicles/gas, and health care in \(1980 .\)

Step-by-Step Solution

Answer

The average per-person spending in 1980 was: $3262 on housing, $2015 on vehicles/gas and $1796 on health care.

Step 1: Define the Variables

Let \( h \) represent the spending on housing, \( v \) represent spending on vehicles/gas, and \( c \) represent the health care spending.

Step 2: Formulate the Equations

The combined spending on housing, vehicles/gas, and health care makes for the first equation: \( h + v+ c = 7073 \). The difference between spending on housing and spending on vehicles/gas gives the second equation: \( h - v = 1247 \). Lastly, the difference between spending on housing and spending on health care gives the third equation: \( h - c= 1466 \).

Step 3: Solve the System of Equations

From the second equation, rearrange for \( v \): \( v = h - 1247 \). Substitute this into the first equation: \( h + (h - 1247) + c = 7073 \). From this, we have a new equation \( 2h + c= 8320 \). Similarly rearrange the third equation for \( c \): \( c = h - 1466 \). Substitute this into the equation just obtained: \( 2h + ( h - 1466) = 8320 \). So, solving this gives \( h = 3262 \). Substitute \( h =3262 \) into the second and third equation to solve for \( v \) and \( c \), respectively. Thus, \( v = 2015 \) and \( c = 1796 \).

Step 4: Interpret the Results

So, the average per-person spending in 1980 was $3262 on housing, $2015 on vehicles/gas and $1796 on health care.

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that focuses on solving systems of linear equations like the one presented in the original exercise. It involves mathematical structures such as vectors and matrices, facilitating their functions and transformations. These concepts are widely applied in various fields, including data science and economics.

In the exercise above, three equations are formulated based on the total and differences in spending between categories. By solving this system of equations, we can find unknown quantities - in this case, the average spending in different areas. To tackle such problems efficiently, linear algebra provides methods such as substitution and elimination.

• **Substitution**: This approach involves isolating one variable and substituting this expression into another equation to progressively reduce the number of equations and variables.


• **Elimination**: Also known as the addition method, it eliminates variables by adding or subtracting equations to zero out a variable, making it simpler to solve for the remaining ones.

Understanding linear algebra is pivotal for solving linear systems especially when more than two variables are involved. It equips students with the mathematical machinery needed to interpret and solve real-world problems like the spending analysis in this exercise.

Spending Analysis

Spending analysis is crucial in understanding and managing finances, whether on a personal, corporate or national level. It involves examining where money is being spent and identifying patterns or areas for optimization. In this exercise, we investigate the average spending in three important categories: housing, vehicles/gas, and healthcare for the year 1980.

The goal was to figure out how much, on average, individuals spent on these essentials. By analyzing the total and differences in spending, we can create a clearer picture of the financial priorities of that era. These insights are essential as they help paint a broader picture of economic conditions and assist in budget planning and policy-making.

• **Housing**: Often the largest expenditure, representing rent or mortgage, utilities, and maintenance costs.
• **Vehicles/Gas**: Includes costs related to transportation, encompassing vehicle maintenance and fuel expenses.


• **Healthcare**: Covers insurance premiums, medication, and other related medical expenses.

By solving the exercise, we understand how these categories consumed the budget and how they potentially impacted individual financial planning in 1980.

Variables in Equations

Variables are symbols or placeholders used in equations to represent quantities that can vary or are yet to be determined. In mathematical terms, a variable can typically be represented by letters such as x, y, or in this case—h, v, and c for housing, vehicles/gas, and healthcare respectively.

Using variables allows us to construct equations that express relationships between these quantities. In our exercise, the equations provided relate the variables to each other and to known quantities, enabling us to solve for the unknowns.

• **Defining Variables**: Identifying what each variable in the system stands for is crucial. Clear definitions reduce confusion and help in logically structuring the equations.
• **Equation Formation**: Once variables are defined, we express the relationships between them through equations. For instance, summing up the expenses gives us one equation, while the difference vectors provide others.

Correctly setting up the variables and equations is key to solving any system. This foundation facilitates the manipulation needed to solve for the unknowns, thereby yielding valuable insights from the exercise.