Problem 60 - College Algebra with Corequisite Support

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. At a market, the three most popular vegetables make up \(53 \%\) of vegetable sales. Corn has \(4 \%\) higher sales than broccoli, which has \(5 \%\) more sales than onions. What percentage does each vegetable have in the market share?

Step-by-Step Solution

Answer

Corn: 22%, Broccoli: 18%, Onions: 13%.

Step 1: Define the Variables

Let the percentage of corn in the market share be \( c \), the percentage of broccoli be \( b \), and the percentage of onions be \( o \). Our goal is to find the values of \( c \), \( b \), and \( o \).

Step 2: Set Up the Equations

From the problem, we know:1. \( c + b + o = 53 \) (total percentage for the three vegetables is 53%)2. \( c = b + 4 \) (corn has 4% higher sales than broccoli)3. \( b = o + 5 \) (broccoli has 5% more sales than onions).We now have a system of three equations.

Step 3: Substitute and Simplify

Use equation 3, \( b = o + 5 \), to substitute \( b \) in the other equations:1. Substitute in equation 1: \( c + (o + 5) + o = 53 \), so \( c + 2o + 5 = 53 \).2. Substitute \( b = o + 5 \) in equation 2 to get \( c = (o + 5) + 4 \) or \( c = o + 9 \).

Step 4: Solve the Equations Applying Cramer's Rule

Now, using the equations from Step 3:1. From \( c + 2o + 5 = 53 \) and \( c = o + 9 \), substitute \( c = o + 9 \) into the first equation: \((o + 9) + 2o + 5 = 53\).2. Simplify: \(3o + 14 = 53\).3. Solve for \( o \): \( 3o = 39 \), so \( o = 13 \).4. Substitute \( o = 13 \) into \( b = o + 5 \): \( b = 18 \).5. Substitute \( o = 13 \) into \( c = o + 9 \): \( c = 22 \).

Step 5: Verify the Solution

Check if these values satisfy the original equation: \( c + b + o = 53 \).Substitute \( c = 22 \), \( b = 18 \), \( o = 13 \):\( 22 + 18 + 13 = 53 \). The values are consistent with the problem statement.

Key Concepts

Cramer's Rule

Cramer's Rule is a handy tool for solving systems of linear equations using determinants. If you have a system of equations with as many equations as unknowns, which is described by a square matrix, Cramer's Rule can give you a direct answer without the need for elaborate computation.
The rule works by calculating the determinant of the coefficient matrix and determinants of matrices obtained by replacing one column of the coefficient matrix with the constant vector.
For each variable in the system, you create a matrix by substituting the column corresponding to the variable with the constants on the other side of the equations. The solution for that variable is then the determinant of this new matrix divided by the determinant of the original coefficient matrix.

• Make sure your system has as many equations as unknowns.
• Cramer's Rule is precise but only efficient for small systems, typically 2x2 or 3x3 matrices.
• Remember the determinant computations can be tedious for larger matrices.

This method offers a straightforward way to find variable values if the conditions are right.

system of equations

A system of equations involves multiple equations that share the same set of variables. In the exercise, we have three equations involving the percentages of corn, broccoli, and onions.
The goal when dealing with a system is to find a set of values for the variables that satisfy all the equations simultaneously.
Systems of equations can be approached using various methods, such as graphing, substitution, elimination, and using matrices like with Cramer's Rule.

• Identify the variables and write equations according to the problem statement.
• Ensure the number of equations matches the number of unknowns for possible unique solutions.
• Consider simplifying or rearranging equations to make solving easier.

Successfully solving a system means that you have found a point (or set of points) where all the conditions of the problem are met.

variable substitution

Variable substitution is a technique used to solve systems by expressing one variable in terms of another, simplifying their interaction within the equations.
In our exercise, substitution aids in making the equations simpler, allowing for direct solving without complicated manipulations.
To effectively apply variable substitution, identify equations where one variable is already isolated.

• Start with the simplest equation to express one variable in terms of others.
• Replace this variable in the other equations to reduce the number of unknowns.
• Simplify the resulting equations until the entire system can be solved easily.

This step-by-step process streamlines solutions by reducing complexity, making it a favored strategy in algebraic problem-solving.

algebraic solution methods

Algebraic solution methods encompass various techniques for solving systems of equations. These may include substitution, elimination, and using matrix operations like Cramer's Rule.
Each method has its strengths and is suited to specific types of problems, emphasizing different aspects of algebraic manipulation and simplification.
Algebraic methods generally prefer symbol manipulation over visual representation, focusing on finding exact answers through a series of logically derived steps.

• Substitution is great for small systems and where equations are easily rearranged.
• Elimination works well if you can add or subtract equations to eliminate a variable.
• Matrix methods, including Cramer's Rule, are powerful for handling multiple variables efficiently.

Choosing the right method often depends on the problem's complexity and the number of equations involved. Understanding these methods allows for flexible problem-solving approaches.