Problem 59 - 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. Your garden produced two types of tomatoes, one green and one red. The red weigh 10 oz, and the green weigh 4 oz. You have 30 tomatoes, and a total weight of \(13 \mathrm{lb}\), 14 oz. How many of each type of tomato do you have?

Step-by-Step Solution

Answer

13 green tomatoes and 17 red tomatoes.

Step 1: Define variables and convert measurements

First, we need to define variables to represent the unknown quantities. Let \( x \) be the number of green tomatoes and \( y \) be the number of red tomatoes. Since 1 pound is 16 ounces, 13 lb 14 oz equals \( 16 \times 13 + 14 = 222 \) oz.

Step 2: Set up the system of equations

Using the problem information, we can set up the following equations:1. \( x + y = 30 \) (since you have a total of 30 tomatoes)2. \( 4x + 10y = 222 \) (since the green tomatoes weigh 4 oz each and the red tomatoes weigh 10 oz each for a total weight of 222 oz).

Step 3: Set up matrices for Cramer's Rule

To use Cramer's Rule, we need to write the system in matrix form:\[\begin{bmatrix}1 & 1 \4 & 10\end{bmatrix}\begin{bmatrix}x \y\end{bmatrix} =\begin{bmatrix}30 \222\end{bmatrix}\]This gives us the coefficient matrix \( A = \begin{bmatrix} 1 & 1 \ 4 & 10 \end{bmatrix} \), the variable matrix \( X = \begin{bmatrix} x \ y \end{bmatrix} \), and the constant matrix \( B = \begin{bmatrix} 30 \ 222 \end{bmatrix} \).

Step 4: Calculate the determinant of the coefficient matrix

The determinant of \( A \) is calculated as:\[ |A| = 1 \cdot 10 - 1 \cdot 4 = 6 \]

Step 5: Calculate determinant for \( x \) (\( D_x \))

Replace the first column of \( A \) with \( B \), resulting in the matrix:\[\begin{bmatrix}30 & 1 \222 & 10\end{bmatrix}\]Calculate the determinant:\[ D_x = 30 \cdot 10 - 1 \cdot 222 = 78 \]

Step 6: Calculate determinant for \( y \) (\( D_y \))

Replace the second column of \( A \) with \( B \), resulting in the matrix:\[\begin{bmatrix}1 & 30 \4 & 222\end{bmatrix}\]Calculate the determinant:\[ D_y = 1 \cdot 222 - 30 \cdot 4 = 102 \]

Step 7: Solve for \( x \) and \( y \) using Cramer's Rule

Using Cramer's Rule:\[ x = \frac{D_x}{|A|} = \frac{78}{6} = 13 \]\[ y = \frac{D_y}{|A|} = \frac{102}{6} = 17 \]

Step 8: Verify the solution

Check the solution by substituting \( x = 13 \) and \( y = 17 \) back into the original equations:1. \( 13 + 17 = 30 \) ✔️2. \( 4 \times 13 + 10 \times 17 = 52 + 170 = 222 \) oz ✔️. This confirms the solution is correct.

Key Concepts

Understanding a System of Linear Equations

In algebra, a system of linear equations consists of two or more linear equations with the same variables. These equations are used to describe relationships between different quantities. For example, let's consider the exercise about determining the number of green and red tomatoes. We use variables to represent unknown quantities, such as the number of green tomatoes as \( x \) and red tomatoes as \( y \). The given conditions provide us with two equations:

• \( x + y = 30 \): This represents the total number of tomatoes.
• \( 4x + 10y = 222 \): This encodes the total weight in ounces of the tomatoes.

These equations form a system that can be solved for the unknowns \( x \) and \( y \). Solving this system requires finding values for the variables that satisfy both equations simultaneously. Often, systems like these are represented in a matrix format to help with solving them using various algebraic methods, such as Cramer's Rule.

The Role of Matrix Determinants

Matrix determinants are incredibly important in linear algebra, especially when dealing with systems of equations. A determinant is a scalar value derived from a square matrix that provides useful insights into the properties of the matrix. For a 2x2 matrix, the determinant can be calculated using the formula \( |A| = ad - bc \) for a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \).In our exercise, the determinant of the coefficient matrix is key to using Cramer's Rule. We calculated it as follows: \[ |A| = 1 \cdot 10 - 1 \cdot 4 = 6 \]This non-zero determinant confirms that the system has a unique solution. If the determinant were zero, it would indicate that the system either has no solution or an infinite number of solutions. That's why the determinant is vital; it ensures the system can be effectively solved using algebraic techniques. Additionally, determinants are used in Cramer's Rule to find individual solutions for each variable by replacing columns in the matrix with the constants from the equations and recalculating the determinant.

Algebraic Problem-Solving with Cramer's Rule

Cramer's Rule is a powerful tool used to solve systems of linear equations with as many equations as variables. It involves using determinants to find the solution for each variable individually. Here’s how it works: 1. Calculate the determinant of the coefficient matrix, which we did as \( |A| = 6 \).2. Replace one column of the coefficient matrix with the constant matrix and calculate the determinant for each variable:

• For \( x \), replace the first column with \( B \): \( D_x = 78 \).
• For \( y \), replace the second column with \( B \): \( D_y = 102 \).

3. Solve for each variable by dividing the appropriate determinant by \( |A| \):

• \( x = \frac{D_x}{|A|} = \frac{78}{6} = 13 \)
• \( y = \frac{D_y}{|A|} = \frac{102}{6} = 17 \)

Cramer's Rule offers a straightforward method for solving such systems, assuming the determinant of the coefficient matrix is non-zero. It converts the algebraic problem-solving process into a series of determinant calculations, providing clear values for each variable. This method is both efficient and effective for systems where the number of equations matches the number of unknowns.