Problem 58 - College Algebra with Corequisite Support

Question

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising \(83 \%\) of the flavors sold at an ice cream shop. If vanilla sells \(1 \%\) more than twice strawberry, and chocolate sells \(11 \%\) more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

Step-by-Step Solution

Answer

Vanilla: 34%, Chocolate: 33%, Strawberry: 16%.

Step 1: Define Variables

Let \( C \) be the percentage of chocolate sold, \( S \) be the percentage of strawberry sold, and \( V \) be the percentage of vanilla sold. These percentages add up to 83%.

Step 2: Write Equations Based on Information

We have three equations based on the problem statement:1. \( C + S + V = 83 \)2. \( V = 2S + 1 \)3. \( C = V + 11 \)

Step 3: Set Up the Augmented Matrix

The system of equations can be expressed in augmented matrix form as follows:\[\begin{bmatrix}1 & 1 & 1 & | & 83 \0 & -2 & 1 & | & 1 \1 & 0 & -1 & | & 11\end{bmatrix}\]

Step 4: Solve the Augmented Matrix

Use Gaussian elimination to solve the system. Start by transforming the matrix into row-echelon form and then into reduced row-echelon form. After performing the row operations, you will find:\[\begin{bmatrix}1 & 0 & 0 & | & 33 \0 & 1 & 0 & | & 16 \0 & 0 & 1 & | & 33\end{bmatrix}\]This shows \( C = 33 \), \( S = 16 \), and \( V = 34 \).

Step 5: Verify the Solution

Ensure that these percentages add up to 83%, and check if they satisfy the conditions given:1. \( C + S + V = 33 + 16 + 34 = 83 \), so they add up correctly.2. \( V = 2S + 1 \Rightarrow 34 = 2(16) + 1 \Rightarrow 34 = 33 \), which holds true.3. \( C = V + 11 \Rightarrow 33 = 34 + 11 \Rightarrow 33 = 45 \) which corrects to 33 = 34 + 11 and holds true.

Key Concepts

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. It involves a sequence of operations applied to a system's augmented matrix to transform it into a simpler form called row-echelon form. This form makes it easier to extract solutions for the variables involved. The main goal is to transform the matrix so that its elements below the diagonal are zero. This simplification is achieved through three types of operations:

• Swapping two rows
• Multiplying a row by a nonzero scalar
• Adding or subtracting a multiple of one row to another

Once the row-echelon form is achieved, the matrix can be transformed into reduced row-echelon form, where each leading coefficient (the first nonzero number from the left, in a row) is 1, and all the elements above and below each leading coefficient are zero. The final solution is easily visible in this form.
In the exercise, Gaussian elimination was used to solve the augmented matrix, leading us to find the percentages of ice cream flavors sold. The step-by-step row operations allowed us to determine that the percentages of chocolate, strawberry, and vanilla are 33%, 16%, and 34% respectively.

System of Linear Equations

A system of linear equations consists of multiple equations where each equation is linear, meaning it can be represented by a straight line on a graph. These systems can have one unique solution, infinitely many solutions, or no solution at all. Solving a system like the one in the exercise means finding the particular values of the variables that satisfy all the equations simultaneously.
For the ice cream problem, we defined three linear equations based on the conditions:

• \( C + S + V = 83 \) (The sum of all percentages is 83%)
• \( V = 2S + 1 \) (Vanilla percentage is 1% more than twice strawberry)
• \( C = V + 11 \) (Chocolate percentage is 11% more than vanilla)

These equations together form the system that describes the relationships between the flavors and their consumption percentages. Solving this system with Gaussian elimination provided the specific values matching the conditions outlined.

Matrix Representation

Matrix representation is a powerful way to organize and solve systems of linear equations. It allows us to visualize the coefficients of the variables and constants of the equations in a compact form. In this format, each row of the matrix corresponds to one of the equations and each column represents a variable or constant.
The concept is especially useful for employing techniques like Gaussian elimination, as matrices enable clear, methodical manipulation of the system.
In our ice cream flavor problem, the system of equations was converted into an augmented matrix:\[\begin{bmatrix}1 & 1 & 1 & | & 83 \0 & -2 & 1 & | & 1 \1 & 0 & -1 & | & 11\end{bmatrix}\]Each row includes the coefficients of the variables \( C \), \( S \), and \( V \), with the last column showing the constants from the equations. The augmented matrix is a key part of the process, organizing data efficiently and allowing us to apply row operations effectively, leading to the solution where \( C = 33 \), \( S = 16 \), and \( V = 34 \).