Problem 77 - College Algebra

Question

The function \(y=0.94 x+5.64\) models annual U.S. consumption of chicken, \(y,\) in pounds per person, \(x\) years after \(1950 .\) The function \(0.74 x+y=146.76\) models annual U.S. consumption of red meat, \(y,\) in pounds per person, \(x\) years after 1950 . What is the most efficient method for solving this system? What does the solution mean in terms of the variables in the functions? (It is not necessary to solve the system.)

Step-by-Step Solution

Answer

The most efficient method to solve this system is substitution and the solution would indicate the year and amount in pounds per person when the U.S annual consumption of both chicken and red meat is the same.

Step 1: Review of linear system solving methods

There are three common methods to solve a system of linear equations: graphing, substitution, and elimination. The most efficient method often depends on the system itself and other requirements of the problem.

Step 2: Choosing the most efficient method

In this case, the most efficient method to solve the system would be substitution because one of the equations is already solved for y, allowing us to substitute \((y = 0.94x + 5.64)\) into the second equation for y.

Step 3: Interpreting the Solution

The solution to this system would give us the year (value of x) when the consumption per person of both chicken and red meat is the same in the U.S. The y value would represent the amount in pounds consumed per person in that specific year.

Key Concepts

Linear Functions

Linear functions are equations that create a straight line when graphed. They have the general form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) represents the change in \( y \) for every unit increase in \( x \). The y-intercept \( b \) is where the line crosses the y-axis. Linear functions are useful for modeling real-world situations where change is constant over time.

• In our exercise, the function \( y = 0.94x + 5.64 \) models chicken consumption per person over time.
• The slope \( 0.94 \) indicates that each year, consumption increases by 0.94 pounds per person.
• The y-intercept \( 5.64 \) suggests that in 1950, the consumption was 5.64 pounds per person.

Understanding these components helps in predicting and analyzing trends in data, as well as solving systems of equations.

Substitution Method

The substitution method is a technique for solving systems of linear equations where one equation is solved for a variable, and this expression is substituted into another equation. This method is especially useful when one of the equations is easily solved for one of the variables.

• An advantage of the substitution method is its straightforward approach in situations where a variable is already isolated.
• In the current problem, the equation \( y = 0.94x + 5.64 \) is solved for \( y \), enabling direct substitution into another equation.
• Substitution reduces systems to a single equation in one variable, simplifying the solution process.

This method is often considered efficient for certain linear systems, such as the one provided in the exercise, because it minimizes calculations and potential errors.

Elimination Method

The elimination method involves manipulating equations in a system to eliminate one of the variables, making it simpler to solve for the other variable. This is done by adding or subtracting equations after possibly multiplying by constants to align the coefficients.

• This method is suitable when the goal is to eliminate variables by combining equations.
• Despite the usefulness of elimination, it wasn't selected in this exercise as one equation was already isolated, making substitution more efficient.
• Elimination can be more complex, requiring careful arithmetic operations to avoid mistakes.

While not used in our current problem, mastering elimination can be particularly helpful when dealing with systems where substitution is less practical.

Graphing Method

Graphing is another approach to solving systems of linear equations. It involves plotting each equation on a graph to find their point of intersection, which represents the solution to the system.

• Graphing provides a visual representation, making it easier to interpret when solutions are simple integers.
• However, with more complex coefficients or solutions, graphing can be less precise than algebraic methods.
• In our problem, graphing wasn’t chosen as the best method because algebraic approaches, like substitution, yield more exact results for non-integer intersections.

Graphing can be immensely useful in understanding the solutions' behavior and in educational settings where visual learning is emphasized.