Question

The equations \(x+2 y+3 z=1,2 x+y+3 z=2\) \(5 \mathrm{x}+5 \mathrm{y}+9 \mathrm{z}=4\) have.... (a) no solution (b) unique solution (c) Infinity solutions (d) cannot say anything

Short answer

The given system of linear equations has a unique solution, based on the Gaussian Elimination method and the resulting row echelon form of the augmented matrix: \( \left[ \begin{array}{ccc|c} 1 & 2 & 3 & 1 \\ 0 & -3 & -3 & 0 \\ 0 & 0 & -1 & -1 \end{array} \right] \). Thus, the correct option is (b) unique solution.

Step-by-step solution

Write the augmented matrix of the linear system

To represent the linear system as an augmented matrix, we will have a 3x4 matrix, where the first three columns represent the coefficients of x, y, and z, and the fourth column represents the constants. \( \left[ \begin{array}{ccc|c} 1 & 2 & 3 & 1 \\ 2 & 1 & 3 & 2 \\ 5 & 5 & 9 & 4 \end{array} \right] \)

Perform Gaussian Elimination to reduce the matrix to row echelon form

Gaussian Elimination involves using row operations to eliminate variables and obtain a triangular matrix. First, eliminate 'x' from the second and third rows by subtracting the first row multiplied by the appropriate scalar: \(R_2 = R_2 - 2R_1\) \(R_3 = R_3 - 5R_1\) \( \left[ \begin{array}{ccc|c} 1 & 2 & 3 & 1 \\ 0 & -3 & -3 & 0 \\ 0 & -5 & -6 & -1 \end{array} \right] \) Next, eliminate 'y' from the third row by subtracting the second row multiplied by the appropriate scalar: \(R_3 = R_3 - \frac{5}{3}R_2\) \( \left[ \begin{array}{ccc|c} 1 & 2 & 3 & 1 \\ 0 & -3 & -3 & 0 \\ 0 & 0 & -1 & -1 \end{array} \right] \)

Analyze the reduced matrix

The given row echelon form of the augmented matrix is: \( \left[ \begin{array}{ccc|c} 1 & 2 & 3 & 1 \\ 0 & -3 & -3 & 0 \\ 0 & 0 & -1 & -1 \end{array} \right] \) Looking at the last row, we can see that 'z' has a unique value, which is z=1. Since 'z' has a unique value, there will be a unique solution for the whole system. Therefore, we choose option (b) unique solution.

Key concepts

Gaussian Elimination

Gaussian Elimination is a powerful algorithm used to solve systems of linear equations. It turns a complex system into a simpler one by performing row operations until the system is in a form that makes the solution apparent. To apply Gaussian Elimination, we typically perform three types of row operations: swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting a multiple of one row from another row. These operations preserve the solutions of the system while systematically removing variables from the equations to find their values step by step. Simplifying a matrix step by step is strategic; it allows us to isolate variables, ultimately leading to quick and accurate solutions. By carefully choosing and applying these row operations, as done in the textbook exercise, students can methodically unravel even the most intricate systems of equations.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It brings together the coefficients of the variables and the constants from each equation into one matrix. In the exercise, we create an augmented matrix that holds the coefficients of the variables, x, y, and z, in the first three columns, and the constants of each equation in the fourth column. This matrix is a valuable tool for Gaussian Elimination because it condenses all the information needed to find the solution into a clear, tabular format. Using augmented matrices streamlines the computation process and minimizes the risk of arithmetic errors, making them an essential aspect of linear algebra.

Row Echelon Form

Row echelon form is a stage in solving linear equations where the augmented matrix has been manipulated so that each row after the first has one fewer non-zero element than the row above it, leading to a triangular form. Characteristics of row echelon form include having all zeros below the leading coefficient (which is the left-most non-zero number) in each row and having the leading coefficient of a lower row positioned to the right of the leading coefficient above it. By achieving row echelon form, as seen in the given solution, we're able to identify pivotal points for each variable, simplifying the process of solving each equation. The goal of Gaussian Elimination is to reach this form, which provides a clear pathway to the solution.

Unique Solution

A system of linear equations has a unique solution when there is exactly one set of values for the variables that satisfies all equations simultaneously. This outcome often results from having as many distinct equations as there are variables, with none of the equations being multiples of each other. The presence of a unique solution is significant since it implies precise and singular answers for each variable, avoiding ambiguity. In the textbook example, the row echelon form indicates that there is a unique solution because the triangular form reflects that each variable can be solved for independently. The presence of a unique solution is important because it ensures that the system is not only solvable but also deterministic, meaning each variable can be found with certainty.