Problem 57 - 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. 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 paint were used.

Step 1: Define Variables

Let \(x\) be the number of gallons of yellow paint, and \(y\) be the number of gallons of blue paint. We need to find values for \(x\) and \(y\).

Step 2: Setup the System of Equations

From the problem, we know the total gallons of paint is 10, and that the total cost is $29.50. We can express these conditions with equations: \(x + y = 10\) (equation for the total gallons) and \(2.59x + 3.19y = 29.50\) (equation for the total cost).

Step 3: Setup Matrices for Cramer's Rule

Identify the coefficient matrix \(A\) and the matrices for \(x\) and \(y\) variables: \[ A = \begin{bmatrix} 1 & 1 \ 2.59 & 3.19 \end{bmatrix}, \] \[ A_x = \begin{bmatrix} 10 & 1 \ 29.50 & 3.19 \end{bmatrix}, \] \[ A_y = \begin{bmatrix} 1 & 10 \ 2.59 & 29.50 \end{bmatrix}. \]

Step 4: Calculate Determinant of Coefficient Matrix

Calculate \(\text{det}(A)\): \[ \text{det}(A) = (1)(3.19) - (1)(2.59) = 3.19 - 2.59 = 0.60. \]

Step 5: Calculate Determinant for x

Calculate \(\text{det}(A_x)\): \[ \text{det}(A_x) = (10)(3.19) - (1)(29.50) = 31.9 - 29.5 = 2.4. \]

Step 6: Calculate Determinant for y

Calculate \(\text{det}(A_y)\): \[ \text{det}(A_y) = (1)(29.50) - (10)(2.59) = 29.5 - 25.9 = 3.6. \]

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

Cramer's rule states that \(x = \frac{\text{det}(A_x)}{\text{det}(A)}\) and \(y = \frac{\text{det}(A_y)}{\text{det}(A)}\). So, \[ x = \frac{2.4}{0.60} = 4, \] \[ y = \frac{3.6}{0.60} = 6. \]

Step 8: Conclusion

The solution is that 4 gallons of yellow paint and 6 gallons of blue paint were used to mix the green paint.

Key Concepts

Cramer's Rule

Cramer's Rule is a handy tool in linear algebra, allowing you to solve a system of linear equations using determinants. It requires that you construct something called a coefficient matrix from the coefficients of the variables in your equations. This rule works only when the number of equations matches the number of variables. Cramer's Rule provides a direct way to find the value of each variable:

• Consider the system of linear equations given as: \[ \begin{align*} a_1x + b_1y &= c_1 \a_2x + b_2y &= c_2 \end{align*} \]
• Create the coefficient matrix using the coefficients of the variables:

\[ A = \begin{bmatrix} a_1 & b_1 \ a_2 & b_2 \end{bmatrix} \] Next, calculate the determinant for the coefficient matrix, for each variable's matrix, and use the formula:

• \( x = \frac{\text{det}(A_x)}{\text{det}(A)} \)
• \( y = \frac{\text{det}(A_y)}{\text{det}(A)} \)

By understanding these steps, Cramer's Rule can simplify solving systems of equations, offering a clear and organized process.

Determinants

A determinant provides valuable information about a matrix, specifically regarding its invertibility. It is a scalar value that can be calculated from a square matrix.To find a determinant of a 2x2 matrix, you can use the simple formula:\[ \text{det}(A) = ad - bc \] where \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Here, each letter represents a number in the matrix itself.Determinants are essential in Cramer's Rule because:

• They help determine whether a system of equations has a unique solution.
• If the determinant of the coefficient matrix is zero, the system doesn't have a unique solution.
• If non-zero, it confirms that we can use Cramer’s Rule for solving.

Understanding the concept of determinants is crucial when working with linear equations and matrices, as it helps in understanding the nature of the solutions.

Matrices

Matrices are fundamental blocks in linear algebra, resembling tables of numbers that represent a set of equations.In our problem, we see several matrices:

• The **coefficient matrix**, which holds the coefficients of the variables.
• **Variable-specific matrices** \( A_x \) and \( A_y \), which are augmented to incorporate different constants from the equations to solve for specific variables.

In linear algebra, matrices help simplify complex systems into a form that is easier to manipulate. Their operations include:

• **Addition** – Combining matrices of the same size.
• **Multiplication** – More involved than addition and requiring specific rules for operations.
• **Transformation** – Applying matrices as transformations, changing coordinates, and much more.

Understanding how to work with matrices helps to unlock more powerful methods and concepts in linear algebra and is useful to solve real-world problems efficiently.

Linear Algebra

Linear Algebra provides a systematic approach for solving systems of linear equations and is a key component of mathematics linked with matrices and determinants. Key aspects of linear algebra include:

• **Vectors** – Objects represented by arrays of numbers, which often define direction and magnitude.
• **Spaces** – Defines the arenas where vectors operate, including two-dimensional and higher-dimensional spaces.
• **Transformations** – Applying operations to vectors and spaces, often described by matrices.
• **Solving Systems** – Finding solutions to sets of linear equations using methods like substitution, elimination, and Cramer's Rule.

Linear Algebra helps in understanding complex systems like those found in physics, engineering, computer science, etc., making it indispensable in these fields. It simplifies complex systems into manageable calculations, providing a pathway from abstract math to practical applications.