Problem 59 - Algebra A Combined Function

Question

Two occupations predicted to greatly increase in the number of jobs are pharmacy technicians and network system analysts. The number of pharmacy technician jobs predicted for 2006 through 2016 can be approximated by \(9.1 x-y=-295 .\) The number of network system analyst jobs predicted for 2006 through 2016 can be approximated by \(14 x-y=-262 .\) For both equations, \(x\) is the number of years since 2006 , and \(y\) is the number of jobs in the thousands. (Source: Bureau of Labor Statistics) a. Use the addition method to solve this system of equations. (Eliminate \(y\) first and solve for \(x\). Round this result to the nearest whole.) b. Interpret your solution from part (a). C. Using the year in your answer to part (b), estimate the number of pharmacy technician jobs and network system analyst jobs in that year.

Step-by-Step Solution

Answer

In 2013, there are approximately 359,000 pharmacy technician jobs and 360,000 network system analyst jobs.

Step 1: Align the Equations

We start by writing both equations in the standard linear form:\[ 9.1x - y = -295 \]\[ 14x - y = -262 \]

Step 2: Eliminate y

To eliminate \(y\), subtract the second equation from the first:\[ (9.1x - y) - (14x - y) = -295 + 262 \]This simplifies to:\[ 9.1x - 14x = -295 + 262 \]\[ -4.9x = -33 \]

Step 3: Solve for x

Divide both sides by -4.9 to find \(x\):\[ x = \frac{-33}{-4.9} \]\[ x \approx 6.73 \]Round \(x\) to the nearest whole number:\[ x = 7 \]

Step 4: Interpret the Value of x

The value \(x = 7\) indicates that 7 years after 2006, which corresponds to the year 2013, the number of jobs in both occupations will be equal.

Step 5: Calculate Pharmacy Technician Jobs

Substitute \(x = 7\) back into the first equation:\[ 9.1 \times 7 - y = -295 \]\[ 63.7 - y = -295 \]\[ y = 63.7 + 295 \]\[ y = 358.7 \]Approximately 359,000 pharmacy technician jobs.

Step 6: Calculate Network System Analyst Jobs

Substitute \(x = 7\) back into the second equation:\[ 14 \times 7 - y = -262 \]\[ 98 - y = -262 \]\[ y = 98 + 262 \]\[ y = 360 \]360,000 network system analyst jobs.

Key Concepts

Addition Method

The addition method is a popular technique used to solve systems of linear equations. It's also known as the elimination method, as it helps eliminate one of the variables to simplify the process.

In the addition method, you manipulate the equations to allow one of the variables to cancel itself out when the equations are combined. Here's how it works:

• Align each equation so that variables and constants are matched in columns. Look at the problem at hand, where we have slightly different coefficients for the variable \(y\).
• Make sure the coefficients of \(y\) are the same in both equations. If they are not, alter one or both equations by multiplying through by a common factor, though in this example, subtraction allows \(y\) to cancel out.

After the \(y\) variable is eliminated, what remains is a simple equation with a single variable, \(x\). This equation is easier to solve, allowing us to find the value for \(x\).

This method is particularly useful when both equations are of similar forms and have easily manageable elements, such as the prediction model for jobs.

Interpretation of Solutions

Once you have determined your values for the variables, it is crucial to interpret what these values mean in the context of the problem. This gives you the real-world application of numbers.

• In this scenario, after using the addition method, we find \(x = 7\).
• This \(x\) value represents the number of years from the base year of 2006.

Calculating this, we add 7 years to 2006, leading us to the year 2013. Hence, the solution tells us that in 2013, the number of jobs in both occupations—pharmacy technicians and network system analysts—will reach a predicted equality point.

The interpretation provides context and meaning, transforming our mathematical results into an understandable answer about job predictions over time.

Linear Equations

Linear equations are fundamental building blocks for modeling relationships between variables. These equations form straight lines when plotted on a graph and are typically written in the form \(ax + by = c\).

In the given problem, our equations are related to job growth: \(9.1x - y = -295\) and \(14x - y = -262\).

• Here, \(x\) represents years since 2006, while \(y\) is the number of jobs in thousands.
• The linear term with \(x\) indicates the growth rate of jobs per year.

Linear equations are crucial because they allow us to predict future behavior based on past trends. Through simple manipulation, they give us insights into how two variables interact, such as predicting job growth in particular careers over time.

The core beauty of linear equations lies in their simplicity and direct applicability to real-world situations.

Job Prediction Models

Job prediction models rely on mathematical equations to forecast future trends in employment. These models use historical data to establish patterns and predict future job opportunities in various sectors.

In the exercise, the model uses linear equations to predict the number of jobs over a decade for pharmacy technicians and network system analysts. Here's why these models are useful:

• They help identify potential job growth areas, which can guide educational institutions in preparing relevant courses and training.
• Employers and policymakers can use them to anticipate workforce needs.
• Individuals can make informed career choices based on predicted job stability and growth.

Job prediction models are an excellent merger of statistical, analytical, and algebraic tools, contributing significantly to strategic decision-making in education, government, and business sectors. Understanding the predictions, as done through linear equations, provides a strategic glimpse into the future.