Problem 24 - Precalculus

Question

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} 2 x-3 y=2 \\ 10 x-15 y=20 \end{array}$$

Step-by-Step Solution

Answer

The determinant is zero, so Cramer's Rule cannot be used.

Step 1: Recognize Cramer's Rule Conditions

Cramer's Rule can only be applied if the number of equations matches the number of variables and if the determinant of the coefficient matrix is non-zero. Here, we have two equations with two unknowns, so we can proceed to check the determinant of the coefficient matrix.

Step 2: Form the Coefficient Matrix

The coefficient matrix for the given system of equations \( 2x - 3y = 2 \) and \( 10x - 15y = 20 \) is\[A = \begin{bmatrix} 2 & -3 \ 10 & -15 \end{bmatrix}\]

Step 3: Calculate the Determinant of the Coefficient Matrix

The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( \text{det}(A) = ad - bc \). For our matrix:\[\text{det}(A) = (2)(-15) - (-3)(10) = -30 + 30 = 0\]

Step 4: Analyze the Determinant

Since the determinant of the coefficient matrix is zero, the system of equations has either no solutions or infinitely many solutions. Cramer's Rule cannot be used as it requires a non-zero determinant.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements and plays a critical role in solving systems of equations using methods such as Cramer's rule. To understand the determinant, think of it as a scalar value that can characterize some properties of the matrix. A key insight is that if the determinant of a matrix is zero, the matrix is said to be singular. This signals specific implications for the system of equations represented by the matrix:

• If the determinant is zero, it may mean that the system has no solutions or infinitely many solutions, as the equations might be dependent.
• If the determinant is non-zero, the system has a unique solution, and methods like Cramer's rule can be applied.

For a 2x2 matrix, the determinant is calculated using the formula: \[\text{det}(A) = ad - bc\]where \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). When using Cramer's rule, always calculate the determinant first to determine if the rule is applicable.

Systems of Equations

Systems of equations are collections of two or more equations that must be solved together, as they share common variables. The primary goal is to find the values of these variables that satisfy all of the equations simultaneously. This concept is foundational in many areas of mathematics and science. Systems of equations can be represented in various forms:

• Linear equations, where variables have exponents of one.
• Non-linear systems that may include squares or higher powers of variables.

When dealing with linear systems, there are several methods to solve them:

• Graphical methods involve plotting each equation on a graph to find the intersection point(s).
• Algebraic techniques like substitution or elimination focus on manipulation to isolate variables.
• Cramer's rule is a technique using the determinant of a matrix, suited for systems with as many equations as variables.

Understanding the structure and solution methods of systems of equations helps in tackling complex real-world problems efficiently.

Coefficient Matrix

A coefficient matrix is a matrix that arises from the coefficients of variables in a system of linear equations. It is an organized way to succinctly represent the constants linked with the variables:Consider the system of equations: \[2x - 3y = 2 \10x - 15y = 20\]The coefficient matrix for this system appears as:\[A = \begin{bmatrix} 2 & -3 \ 10 & -15 \end{bmatrix}\]Each entry in the coefficient matrix corresponds directly to the coefficients of the variables in the original equations. The position of each number matters, as it aligns with the placement of variables \( x \) and \( y \) in each equation.When solving systems of equations using matrix methods like Cramer's rule, the coefficient matrix forms the backbone of calculations. Specifically, the determinant of this matrix is crucial because:

• A non-zero determinant indicates that the system has a unique solution.
• A zero determinant suggests either no solutions or infinitely many solutions.

This makes the coefficient matrix a vital component in addressing linear systems.