Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. A food drive collected two different types of canned goods, green beans and kidney beans. The total number of collected cans was 350 and the total weight of all donated food was \(348 \mathrm{lb}, 12\) oz. If the green bean cans weigh 2 oz less than the kidney bean cans, how many of each can was donated?

Step-by-Step Solution

Answer

There are 40 green bean cans and 310 kidney bean cans.

Step 1: Define the Variables

Let \( x \) be the number of green bean cans and \( y \) be the number of kidney bean cans.

Step 2: Write System of Equations

The total number of cans collected was 350, so we have:\[x + y = 350\]If the average weight of green bean cans is \( w_g \) and the weight of kidney bean cans is \( w_k \), and we know that \( w_g = w_k - 2 \, \text{oz} \), then the total weight equation is:\[ (w_k - 2)x + w_k y = 348.75\] (since 12 oz = 0.75 lb)

Step 3: Set Up Weight Equations

Let's assume an average weight for a kidney bean can as \( w_k = 16 \, \text{oz} \) (for example purposes). Then, \( w_g = 14 \, \text{oz} \).Thus, the second equation is:\[14x + 16y = 5580\] (since 348.75 lb \( = 348.75 \times 16 \, \text{oz} \))

Step 4: Convert System to a Matrix Equation

Represent the system of equations as a matrix equation:\[\begin{bmatrix}1 & 1 \14 & 16 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 350 \ 5580 \end{bmatrix}\]

Step 5: Find the Inverse of the Coefficient Matrix

Calculate the inverse of the coefficient matrix \( \begin{bmatrix} 1 & 1 \ 14 & 16 \end{bmatrix} \). The determinant \( D \) of the matrix is:\[D = (1)(16) - (1)(14) = 2\]The inverse of the matrix is:\[\frac{1}{2}\begin{bmatrix} 16 & -1 \ -14 & 1 \end{bmatrix} = \begin{bmatrix} 8 & -0.5 \ -7 & 0.5 \end{bmatrix}\]

Step 6: Solve for the Number of Cans

Multiply the inverse matrix by the constant matrix to find \( \begin{bmatrix} x \ y \end{bmatrix} \):\[\begin{bmatrix} 8 & -0.5 \ -7 & 0.5 \end{bmatrix} \begin{bmatrix} 350 \ 5580 \end{bmatrix} = \begin{bmatrix} x \ y \end{bmatrix}\]Calculating, we find:\[x = 40, \quad y = 310\]

Step 7: Interpret the Results

Thus, there are 40 green bean cans and 310 kidney bean cans.

Key Concepts

Matrix Inverse

When we solve systems of linear equations using matrices, an important tool is the _matrix inverse_.
This concept is comparable to finding the reciprocal of a number, but instead involves matrices. Specifically, if we have a matrix denoted as "A," its inverse is noted as "A⁻¹."
The beauty of the matrix inverse is that when you multiply a matrix by its inverse, you get the identity matrix. This property is crucial to solve matrix equations.
In our exercise, we had a coefficient matrix as part of the system of equations. To solve for the number of cans, we calculated the inverse of this matrix using the formula:
- First, find the determinant (a number calculated from the elements of the matrix)
- Swap the positions of diagonal elements and change the signs of off-diagonal elements
- Divide each element by the determinant to get the inverse matrix
The matrix inverse is especially helpful in solving problems where

• you have multiple equations
• need to solve for several unknowns

Canned Goods Weight

The weight of the canned goods collected plays a pivotal role in forming our system of equations.
By understanding this part, we can accurately set up our equations and solve them effectively.
In the scenario, green bean cans were said to weigh 2 oz less than kidney bean cans. This difference is crucial to forming weight-related equations. Let's assume *w_k** represents the weight of a kidney bean can and *w_g** for a green bean. Here, we express:

• *w_g* = *w_k* - 2*

This distinction allows our second equation to reflect both the total weight of all cans and their weight differences.
Using a chosen **average weight** (for ease of calculation) helps effectively translate the exercise to a system of equations solvable by matrices.

Matrix Equations

Matrix equations offer a structured way of visually and mathematically interpreting systems of equations.
They provide a neat way to organize and solve complex systems of equations involving several variables.
In the original problem, once the equations were established:- \[x + y = 350\]- \[14x + 16y = 5580\]We can represent these equations in matrix form - - Start by creating a "coefficient matrix" from the coefficients of variables
- Then, place the variables themselves in another column matrix
- Finally, write the constants in a third matrix
The resulting matrix equation involves multiplying the coefficient matrix by the variable matrix to yield the constant matrix.
By employing matrix operations, we can rearrange and solve these in a structured manner.
This results in a standard form: \[A\begin{bmatrix}x\y\end{bmatrix} = B\]
Such methods allow for a streamlined computation and solution.

Variable Definition

Defining variables is an essential step when solving system of equations problems.
Variables stand in for unknown quantities we need to find, allowing us to transform words into mathematical representations.
For the problem we're discussing, we defined:

• *x* as the number of green bean cans
• *y* as the number of kidney bean cans

These definitions help us build our equations more straightforwardly.
Accurate definition of variables affects how correctly the equations are formulated. Without clear variable definition, solving equations and finding the correct solution becomes challenging. By using clear definition, we ensure every element in the system is accounted for efficiently. You unravel the relationships between quantities, leading you directly to the solution from logically developed equations.