Problem 59 - College Algebra

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 lb, 14 oz. How many of each type of tomato do you have?

Step-by-Step Solution

Answer

There are 17 red tomatoes and 13 green tomatoes.

Step 1: Define Variables

Let \( r \) represent the number of red tomatoes and \( g \) represent the number of green tomatoes. We want to find \( r \) and \( g \).

Step 2: Establish the System of Equations

We have two pieces of information:1. The total number of tomatoes is 30: \( r + g = 30 \).2. The total weight in ounces is 222 (since 13 lb 14 oz = (13 x 16) + 14 = 222). The weight equation is: \( 10r + 4g = 222 \).

Step 3: Construct the Coefficient Matrix and Solve using Cramer's Rule

The system of equations is:\[\begin{cases} r + g = 30 \ 10r + 4g = 222\end{cases}\]The coefficient matrix is:\[A = \begin{bmatrix} 1 & 1 \ 10 & 4 \end{bmatrix}\]The determinant of \( A \), \( \text{det}(A) \), is:\( \text{det}(A) = (1)(4) - (1)(10) = 4 - 10 = -6 \).

Step 4: Calculate Determinants for Variables

For \( r \): Replace the first column of \( A \) with the constants vector:\[A_r = \begin{bmatrix} 30 & 1 \ 222 & 4 \end{bmatrix}\]Compute \( \text{det}(A_r) = (30)(4) - (1)(222) = 120 - 222 = -102 \).For \( g \): Replace the second column of \( A \) with the constants vector:\[A_g = \begin{bmatrix} 1 & 30 \ 10 & 222 \end{bmatrix}\]Compute \( \text{det}(A_g) = (1)(222) - (30)(10) = 222 - 300 = -78 \).

Step 5: Solve for Variables using Cramer's Rule

Using Cramer's Rule:\( r = \frac{\text{det}(A_r)}{\text{det}(A)} = \frac{-102}{-6} = 17 \)\( g = \frac{\text{det}(A_g)}{\text{det}(A)} = \frac{-78}{-6} = 13 \)

Step 6: Verify the Solution

Substitute \( r = 17 \) and \( g = 13 \) back into the original equations:- Total number of tomatoes: \( r + g = 17 + 13 = 30 \)- Total weight: \( 10(17) + 4(13) = 170 + 52 = 222 \) ozBoth conditions are satisfied, so the solution is correct.

Key Concepts

Understanding System of Linear Equations

A system of linear equations consists of two or more equations that have common variables. These equations describe relationships involving these shared variables.
The main goal in solving a system of equations is to find values for the variables that satisfy all equations at the same time.
In our exercise, the variables are the number of red and green tomatoes, represented by \( r \) and \( g \).
We form two equations based on the given conditions:

• The first equation is \( r + g = 30 \), representing the total number of tomatoes.
• The second equation is \( 10r + 4g = 222 \), representing the total weight of the tomatoes in ounces.

By solving these equations simultaneously, we can determine the exact number of each type of tomato.

What are Determinants?

Determinants are a special set of numbers that can be calculated from the elements of a square matrix. They are very useful in many areas of mathematics, including solving systems of linear equations using Cramer's Rule.
For a 2x2 matrix, the determinant is found by multiplying the diagonals and then subtracting them: if the matrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \).
In this exercise, the matrices \( A \), \( A_r \), and \( A_g \) each have determinants that help us use Cramer's Rule effectively.

• For the coefficient matrix \( A \), the determinant is \( 4 - 10 = -6 \).
• For matrix \( A_r \), the determinant is \( 120 - 222 = -102 \).
• For matrix \( A_g \), the determinant is \( 222 - 300 = -78 \).

These values are crucial to finding the solutions \( r \) and \( g \).

Defining and Solving for Variables

Variables in mathematics allow us to create general equations that embody real-world problems. In this exercise, we use the variables \( r \) and \( g \) to represent the number of red and green tomatoes, respectively.
To solve for these variables using Cramer's Rule, we manipulate the determinant calculations:

• For \( r \), we calculate \( r = \frac{-102}{-6} = 17 \).
• For \( g \), the calculation is \( g = \frac{-78}{-6} = 13 \).

These calculations give us the exact number of red and green tomatoes that correspond to the conditions given in the problem, ensuring that both the total count and weight are satisfied.

Verification of the Solution

Once a solution is found, verification ensures it satisfies the original problem conditions. Verifying solutions is crucial because even a small mistake in calculations can lead to incorrect results.
Here, we substitute \( r = 17 \) and \( g = 13 \) back into the original equations:

• For the total number of tomatoes, check: \( 17 + 13 = 30 \).
• For the total weight in ounces, check: \( 10(17) + 4(13) = 170 + 52 = 222 \) oz.

Both conditions match the problem's criteria, confirming that the solution is accurate and satisfies the system of equations.