Problem 51 - Precalculus

Question

Apply Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} x+y+z=9 \\ x-y+z=3 \\ -x+y-z=5 \end{array}$$

Step-by-Step Solution

Answer

The system has no unique solution because the determinant of the coefficient matrix is zero.

Step 1: Write the System in Matrix Form

First, express the system of equations in matrix form. The system is: \[ \begin{align*} x + y + z &= 9 \ x - y + z &= 3 \ -x + y - z &= 5 \end{align*} \]The coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\) are:\[ A = \begin{bmatrix} 1 & 1 & 1 \ 1 & -1 & 1 \ -1 & 1 & -1 \end{bmatrix}, \ X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \ B = \begin{bmatrix} 9 \ 3 \ 5 \end{bmatrix} \] Thus, the equation is \(AX = B\).

Step 2: Find Determinant of Matrix A

Find the determinant of the matrix \(A\):\[ \det(A) = \begin{vmatrix} 1 & 1 & 1 \ 1 & -1 & 1 \ -1 & 1 & -1 \end{vmatrix} \]Using cofactor expansion, calculate:\[ \det(A) = 1 \cdot \begin{vmatrix} -1 & 1 \ 1 & -1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 1 \ -1 & -1 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & -1 \ -1 & 1 \end{vmatrix} \]Calculate each 2x2 determinant:\[ = 1((-1)(-1) - (1)(1)) - 1((1)(-1) - (1)(-1)) + 1((1)(1) - (-1)(-1)) \]\[ = 1(1-1) - 1(-1+1) + 1(1-1) \]\[ = 0 + 0 + 0 = 0 \]. As \( \det(A) = 0 \), the system does not have a unique solution.

Key Concepts

Understanding Systems of Equations

A system of equations is a collection of two or more equations with a common set of variables. This means that in order to solve such systems, we are looking for values for the variables that satisfy all the equations simultaneously. Every equation in the system provides a condition that the variables must meet. For instance, in the given equations:

• Equation 1: \( x + y + z = 9 \)
• Equation 2: \( x - y + z = 3 \)
• Equation 3: \( -x + y - z = 5 \)

The goal is to find a common solution \((x, y, z)\) that makes each equation true. Solving systems of equations is important in various fields, like engineering, physics, and economics, as these problems often model real-world situations.

Transforming Systems into Matrix Form

To efficiently solve systems of equations, they are often transformed into matrix form. This shifts the problem into linear algebra territory, where powerful mathematical tools can be applied.Let's break down the transition of the given system into matrix form:- The coefficients of the variables \( (x, y, z) \) in the equations form the **coefficient matrix**, denoted as \( A \): \[ A = \begin{bmatrix} 1 & 1 & 1 \ 1 & -1 & 1 \ -1 & 1 & -1 \end{bmatrix} \]- The variables \( x, y, z \) themselves are represented as a **variable matrix** or vector \( X \): \[ X = \begin{bmatrix} x \ y \ z \end{bmatrix} \]- The constants from the equations form the **constant matrix** or vector \( B \): \[ B = \begin{bmatrix} 9 \ 3 \ 5 \end{bmatrix} \]By setting up the equation \( AX = B \), the system of equations is now in matrix form. This allows us to apply methods like Cramer's Rule or matrix inversion if applicable, to find solutions.

Determinant Calculation and Its Importance

Calculating the determinant of a matrix is a key process in determining the nature of solutions for a system of equations. The determinant is a single number that is computed from the elements of a square matrix.For the matrix \( A \), with determinant denoted as \( \det(A) \), the process involves:- Selecting any row or column for cofactor expansion.- Breaking down the calculation step-by-step using smaller 2x2 determinants.For our specific system, we have:\[\det(A) = \begin{vmatrix} 1 & 1 & 1 \ 1 & -1 & 1 \ -1 & 1 & -1 \end{vmatrix} = 1(((-1) \times (-1)) - (1 \times 1)) - 1((1 \times -1) - (-1 \times 1)) + 1((1 \times 1) - (-1 \times -1))\]Calculating yields \( \det(A) = 0 \).When \( \det(A) = 0 \), it indicates there is either no solution or infinitely many solutions to the system, as the matrix is singular. This means it cannot be inverted and unique solutions do not exist using Cramer's Rule. This highlights the importance of determinant calculations in assessing the solvability of systems of equations.