Problem 57 - 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. You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were 10 gal total. Additionally, you kept your receipt, and know the total amount spent was \(\$ 29.50 .\) If each gallon of yellow costs \(\$ 2.59,\) and each gallon of blue costs \(\$ 3.19,\) how many gallons of each color go into your green mix?

Step-by-Step Solution

Answer

4 gallons of yellow and 6 gallons of blue.

Step 1: Define Variables

Let's define our variables. Let \( x \) represent the number of gallons of yellow paint used, and \( y \) represent the number of gallons of blue paint used.

Step 2: Set Up Equations

We know that the total number of gallons used is 10, which gives us the equation:\[ x + y = 10 \]We also know the total cost of the paint is \( \\(29.50 \). Given that yellow paint costs \( \\)2.59 \) per gallon and blue paint costs \( \$3.19 \) per gallon, we have:\[ 2.59x + 3.19y = 29.50 \]

Step 3: Write System of Equations

The system of equations based on our problem is:1. \( x + y = 10 \)2. \( 2.59x + 3.19y = 29.50 \)

Step 4: Set Up Matrices for Cramer's Rule

For Cramer's Rule, define matrices as follows:Matrix \( A \) (coefficients of variables):\[ A = \begin{bmatrix} 1 & 1 \ 2.59 & 3.19 \end{bmatrix} \]Matrix \( B \) (right-hand side constants):\[ B = \begin{bmatrix} 10 \ 29.50 \end{bmatrix} \]

Step 5: Calculate Determinant of A

The determinant of matrix \( A \) is given by:\[ \text{det}(A) = (1)(3.19) - (1)(2.59) = 3.19 - 2.59 = 0.6 \]

Step 6: Calculate Determinant of A with B for x

Replace the first column of \( A \) with \( B \) to find \( \text{det}(A_x) \):\[ A_x = \begin{bmatrix} 10 & 1 \ 29.50 & 3.19 \end{bmatrix} \]\[ \text{det}(A_x) = (10)(3.19) - (1)(29.50) = 31.9 - 29.5 = 2.4 \]

Step 7: Calculate Determinant of A with B for y

Replace the second column of \( A \) with \( B \) to find \( \text{det}(A_y) \):\[ A_y = \begin{bmatrix} 1 & 10 \ 2.59 & 29.50 \end{bmatrix} \]\[ \text{det}(A_y) = (1)(29.50) - (10)(2.59) = 29.5 - 25.9 = 3.6 \]

Step 8: Solve for Variables x and y Using Cramer's Rule

Using Cramer's Rule, the values of \( x \) and \( y \) are:\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{2.4}{0.6} = 4 \]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{3.6}{0.6} = 6 \]

Step 9: Conclusion

Therefore, the solution to the system is that 4 gallons of yellow paint and 6 gallons of blue paint were used in the mix.

Key Concepts

System of Linear Equations

To solve the problem of determining how many gallons of yellow and blue paint were used, we start by forming a system of linear equations. This technique allows us to express complex real-life situations mathematically, which aids in finding solutions efficiently.

In this scenario, the system of linear equations captures two essential pieces of information: the total paint used and the total cost of the paint.

• The first equation captures the total amount of paint, defined as: \( x + y = 10 \) where \( x \) represents the gallons of yellow paint, and \( y \) represents the gallons of blue paint.
• The second equation represents the cost equation: \( 2.59x + 3.19y = 29.50 \), where each term combines the price per gallon of each paint type and aggregates to the total cost.

These equations together form the basis for Cramer's Rule, which is used to solve for the unknowns \( x \) and \( y \).

Determinant Calculation

Determinant calculation is a key component in applying Cramer's Rule, which is a powerful method for solving systems of linear equations. It involves finding the determinant of a matrix, which is a scalar value that can indicate many properties about the matrix.

Specifically, in the context of our system:

• You first calculate the determinant of matrix \( A \), derived from the coefficients of the variables in the equations: \[ A = \begin{bmatrix} 1 & 1 \ 2.59 & 3.19 \end{bmatrix} \] This leads to \( \text{det}(A) = 0.6 \).
• Next, compute \( \text{det}(A_x) \) by replacing the first column of \( A \) with the constants from matrix \( B \), resulting in \( \text{det}(A_x) = 2.4 \).
• Similarly, for \( \text{det}(A_y) \), replace the second column of \( A \) with \( B \), yielding \( \text{det}(A_y) = 3.6 \).

These determinants are vital as they enable the computation of each variable in the system using division rules of Cramer's formula.

Variables Definition

Defining the variables correctly is a crucial step in forming an accurate system of linear equations. It sets the foundation on which the entire solution is built.

In the paint mixing scenario:

• Let \( x \) be the variable representing the number of gallons of yellow paint used. This helps isolate the influence of yellow paint in both volume and expense.
• Let \( y \) be the variable for the number of gallons of blue paint. Similarly, it captures the distinct contribution of blue paint.

Properly defining these variables makes the subsequent steps of setting up equations, arranging matrices, and applying Cramer's Rule straightforward and logical.

Matrix Setup

Matrix setup is essential for using Cramer's Rule. This involves organizing the coefficients and constants into matrices to facilitate determinant calculation and equation solving.

For the problem:

• Matrix \( A \) is organized from the coefficients of the variables in the equations: \[ A = \begin{bmatrix} 1 & 1 \ 2.59 & 3.19 \end{bmatrix} \]
• Matrix \( B \), on the other hand, consists of constants on the right side of the equations: \[ B = \begin{bmatrix} 10 \ 29.50 \end{bmatrix} \]

By setting up the matrices this way, you pave the way for calculating determinants and subsequently applying Cramer's formula quickly for solving both \( x \) and \( y \). Matrix setup transforms the equations from a verbal format into a readily calculable structure.