Problem 63 - A Graphical Approach to College Algebra

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{r}2 x-3 y=-5 \\\x+5 y=17\end{array}$$

Step-by-Step Solution

Answer

The solution is \( x = 2 \) and \( y = 3 \).

Step 1: Write the system in matrix form

The given system of equations is: \( 2x - 3y = -5 \) and \( x + 5y = 17 \). To solve with Cramer's rule, first write the system in matrix form: \( AX = B \). Where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix. Thus, \( A = \begin{bmatrix} 2 & -3 \ 1 & 5 \end{bmatrix} \), \( X = \begin{bmatrix} x \ y \end{bmatrix} \), \( B = \begin{bmatrix} -5 \ 17 \end{bmatrix} \).

Step 2: Calculate the determinant of matrix A

Use the formula for the determinant of a 2x2 matrix: \( D = ad - bc \), where matrix \( A \) is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} 2 & -3 \ 1 & 5 \end{bmatrix} \). Thus, \( D = 2 \cdot 5 - (-3) \cdot 1 = 10 + 3 = 13 \). Since \( D eq 0 \), we can use Cramer's rule.

Step 3: Calculate determinant of modified matrices

Create modified matrices \( A_x \) and \( A_y \) by replacing columns in \( A \) with \( B \), then compute their determinants. \( A_x = \begin{bmatrix} -5 & -3 \ 17 & 5 \end{bmatrix} \) yields determinant \( D_x = (-5)\cdot5 - (-3)\cdot17 = -25 + 51 = 26 \). \( A_y = \begin{bmatrix} 2 & -5 \ 1 & 17 \end{bmatrix} \) yields determinant \( D_y = 2\cdot17 - (-5)\cdot1 = 34 + 5 = 39 \).

Step 4: Apply Cramer's Rule to find values of x and y

Cramer's rule allows us to find \( x \) and \( y \) as follows: \( x = \frac{D_x}{D} \) and \( y = \frac{D_y}{D} \). Plugging in the calculated determinants, we get \( x = \frac{26}{13} = 2 \) and \( y = \frac{39}{13} = 3 \).

Key Concepts

Matrix Determinant

A matrix determinant is a special number that can be calculated from a square matrix. It is essential in solving systems of linear equations, finding inverse matrices, and determining solutions using Cramer's Rule. In the context of a 2x2 matrix, the determinant provides insight into the matrix's properties and whether a unique solution for the system exists.
To calculate the determinant of a 2x2 matrix, represented as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula is:

• \( D = ad - bc \)

This expression essentially multiplies the diagonal elements and subtracts the product of the off-diagonal elements.
If the determinant (\( D \)) is zero, the matrix does not have an inverse, indicating that the linear equations might be dependent or might not have a unique solution. Conversely, if \( D eq 0 \), as in the example given where \( D = 13 \), it suggests that the system of equations is independent and can be solved with certainty using methods like Cramer's Rule.

System of Equations

A system of equations consists of multiple equations that share the same variables. Solving such a system means finding the values of the variables that satisfy all the equations simultaneously. Systems can be either consistent, inconsistent, or dependent:

• Consistent: One or more solutions exist.
• Inconsistent: No solutions exist as the equations contradict each other.
• Dependent: Infinite solutions exist as all the equations describe the same plane or line.

In the example provided, the equations are: * \( 2x - 3y = -5 \)* \( x + 5y = 17 \)These two equations need to be satisfied at the same time. Using Cramer's Rule is one way to do this when the determinant of the coefficient matrix is non-zero. The determinant indicates whether the equations form a consistent and independent system. In this example, the non-zero determinant confirms the relationships are independent, and unique solutions (\( x = 2 \), \( y = 3 \)) exist that satisfy both equations.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vector spaces, linear mappings, and systems of linear equations.
It is foundational for understanding various concepts in mathematics, physics, engineering, computer science, and economics. One of the key components in linear algebra is the study of matrices and determinants, which are crucial in solving systems of linear equations using techniques like Cramer's Rule.
Here are some core ideas in linear algebra that relate to solving systems of equations:

• Vectors and Matrices: Vectors are quantities with magnitude and direction, while matrices are rectangular arrays of numbers that represent linear transformations.
• Linear Transformations: These are mappings between vector spaces that preserve vector addition and scalar multiplication.
• Matrix Operations: Includes addition, subtraction, multiplication, and finding inverses, which are fundamental in manipulating equations.
• Determinants: As previously discussed, determinants are used to assess the properties of matrices, such as invertibility, which is crucial for Cramer's Rule.
• Rank and Independence: Determining the rank of a matrix helps identify independent equations or vectors essential for understanding the solutions' nature.

Understanding these concepts is indispensable for solving systems of equations efficiently, especially in more complex scenarios often encountered in advanced studies of linear algebra.