Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{l} 4 x+5 y=13 \\ 3 x+y=-4 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = -3 \) and \( y = 5 \).

Step 1: Write Down the System of Equations

The given system of equations is: \[ \begin{align*} 4x + 5y &= 13 \ 3x + y &= -4 \end{align*} \]

Step 2: Check Applicability of Cramer's Rule

Cramer's Rule can be applied if the determinant of the coefficient matrix is non-zero. Here, the coefficient matrix is \[ A = \begin{bmatrix} 4 & 5 \ 3 & 1 \end{bmatrix} \] with a determinant \( \det(A) = (4 \times 1) - (5 \times 3) = 4 - 15 = -11 \). Since the determinant is not zero, Cramer's Rule is applicable.

Step 3: Solve for x Using Cramer's Rule

To solve for \( x \), replace the first column of the coefficient matrix with the constants from the right side of the equations, resulting in the matrix \[ A_x = \begin{bmatrix} 13 & 5 \ -4 & 1 \end{bmatrix} \]. Calculate its determinant: \[ \det(A_x) = (13 \times 1) - (5 \times -4) = 13 + 20 = 33 \]. Then, \( x = \frac{\det(A_x)}{\det(A)} = \frac{33}{-11} = -3 \).

Step 4: Solve for y Using Cramer's Rule

To solve for \( y \), replace the second column of the coefficient matrix with the constants, resulting in the matrix \[ A_y = \begin{bmatrix} 4 & 13 \ 3 & -4 \end{bmatrix} \]. Calculate its determinant: \[ \det(A_y) = (4 \times -4) - (13 \times 3) = -16 - 39 = -55 \]. Then, \( y = \frac{\det(A_y)}{\det(A)} = \frac{-55}{-11} = 5 \).

Step 5: Write the Solution

The solution to the system of equations using Cramer's Rule is \( x = -3 \) and \( y = 5 \).

Key Concepts

Determinant

In linear algebra, the determinant is a special number that can be calculated from a square matrix. Determinants provide insights into the properties of the matrix, such as whether it has an inverse and the characteristics of solutions to the associated systems of equations.

To calculate the determinant of a 2x2 matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]we use the formula:\[\det(A) = ad - bc.\]The determinant is essential for determining whether Cramer's Rule can be applied. If the determinant is non-zero, the matrix has an inverse, and the system of equations has a unique solution.

• Zero Determinant: Indicates that the matrix does not have an inverse and the system may have no solutions or infinitely many.
• Non-zero Determinant: Means the system has a unique solution, which makes Cramer's Rule applicable.

In our example, the determinant for the coefficient matrix was -11, meaning Cramer's Rule could be used.

System of Equations

A system of equations consists of two or more equations that share common variables. The goal is to find values for these variables that satisfy all equations simultaneously. For example, in the given system:
\[\begin{align*} 4x + 5y &= 13 \ 3x + y &= -4 \\end{align*}\]the variables are \(x\) and \(y\).

• Consistent System: A system that has at least one solution.
• Inconsistent System: A system that has no solution.
• Dependent System: When equations describe the same line, leading to infinitely many solutions.

In our exercise, the system had a unique solution. Linear algebra methods, such as Cramer's Rule, help efficiently solve these systems when applicable.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It provides the tools for solving linear systems of equations, a common task in engineering and science.

Key concepts include:

• Vectors and Vector Spaces: Essential building blocks that represent quantities defined by both magnitude and direction.
• Matrices: Rectangular arrays of numbers that can represent systems of equations. Operations include addition, subtraction, multiplication, and finding inverses.
• Linear Transformations: Functions that map vectors to vectors in a way that preserves vector addition and scalar multiplication.
• Determinants and Eigenvalues: Determinants help determine if a matrix is invertible, while eigenvalues provide insights into matrix characteristics.

Linear algebra is foundational in understanding and solving linear equations, such as those encountered when using methods like Cramer's Rule.