Problem 23 - Precalculus

Question

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{rr} 2 x-3 y= & 4 \\ -10 x+15 y= & -20 \end{array}$$

Step-by-Step Solution

Answer

The system cannot be solved using Cramer's rule as the determinant is zero, indicating either no solutions or infinitely many solutions.

Step 1: Identify Coefficients

First, write down the given system of equations and identify the coefficients of each variable: 1. \(2x - 3y = 4\)2. \(-10x + 15y = -20\).Coefficients:- \(a_1 = 2\), \(b_1 = -3\), \(c_1 = 4\)- \(a_2 = -10\), \(b_2 = 15\), \(c_2 = -20\).

Step 2: Set Up Determinant of Coefficient Matrix

Calculate the determinant of the coefficient matrix (denoted as \(D\)) using the formula:\[ D = \begin{vmatrix} a_1 & b_1 \ a_2 & b_2 \end{vmatrix} = \begin{vmatrix} 2 & -3 \ -10 & 15 \end{vmatrix} \]\(D = (2)(15) - (-3)(-10) = 30 - 30 = 0\).

Step 3: Determine System Solution Possibility

Since \(D = 0\), Cramer's Rule cannot be used because it indicates that the system has either no solutions or infinitely many solutions. This scenario suggests the two equations are dependent, meaning they represent the same line and thus intersect at infinitely many points.

Key Concepts

Understanding Determinant

The determinant of a matrix is a special value that is calculated from its elements. It's an important concept, especially when solving systems of linear equations using methods like Cramer's Rule. In the context of a 2x2 matrix, the determinant can be visualized as the area of a parallelogram formed by its vectors. This geometric interpretation helps in understanding why a zero determinant implies that the vectors (or lines) are on top of each other, hence dependent.

The formula for the determinant of a 2x2 matrix is:

• \(D = \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc\)

For example, the matrix derived from the coefficients of the given system:

• \( \begin{vmatrix} 2 & -3 \ -10 & 15 \end{vmatrix}\)

calculates as \((2)(15) - (-3)(-10) = 0\).

When the determinant is zero, it usually suggests the lines are parallel or, as in this case of dependency, they overlap completely.

Exploring System of Equations

A system of equations involves solving multiple equations together, where you find values of variables that satisfy all equations simultaneously. These systems are common in algebra and are used to solve real-world problems where multiple conditions must be met at the same time. Typically, a system may have:

• Exactly one solution
• No solution
• Infinitely many solutions

The given system in our problem consists of two linear equations:

• \(2x - 3y = 4\)
• \(-10x + 15y = -20\)

In these equations, the solution represents points (values of \(x\) and \(y\)) that lie on both lines. When using methods like Cramer's Rule to solve such systems, the determinant of the coefficient matrix often informs us about the nature of the solutions. A non-zero determinant typically means a unique solution exists.

Understanding Dependent Equations

Dependent equations are those where one equation can be derived from another by scaling or some algebraic manipulation. In the context of our original exercise, both given equations are essentially the same, just scaled differently. This is evident from the zero determinant, which shows that these equations correspond to the same line.Dependent systems result in infinitely many solutions because every point on the line described by the equation will satisfy both equations. This relationship tells us that instead of crossing at a single unique point, these lines coincide.

So in simple terms:

• If you express the second equation as \(-5\) times the first, you realize they are just different forms of the same equation. This demonstrates dependency.

Often, dependent systems arise naturally when the conditions you're trying to solve describe the same situation, just expressed differently in terms of scale or orientation. Understanding this can simplify solving many practical algebra problems.