Question

Convert the given system of linear equations into an augmented matrix. $$ \begin{array}{l} 3 x+4 y+5 z=7 \\ -x+y-3 z=1 \\ 2 x-2 y+3 z=5 \end{array} $$

Short answer

The augmented matrix is \[\begin{bmatrix} 3 & 4 & 5 & | & 7 \\ -1 & 1 & -3 & | & 1 \\ 2 & -2 & 3 & | & 5 \end{bmatrix}\].

Step-by-step solution

Identify the coefficients and constants

Firstly, look at each equation in the system individually. For the first equation, \(3x + 4y + 5z = 7\), the coefficients for \(x\), \(y\), and \(z\) are 3, 4, and 5, respectively, and the constant is 7. Similarly, for the second equation \(-x + y - 3z = 1\), the coefficients are -1, 1, and -3, with a constant of 1. For the third equation \(2x - 2y + 3z = 5\), the coefficients are 2, -2, and 3, with a constant of 5.

Construct the Augmented Matrix

Using the coefficients and constants identified, form the augmented matrix. Each row in the matrix represents the coefficients from one equation, followed by the constant term:\[\begin{bmatrix}3 & 4 & 5 & | & 7 \-1 & 1 & -3 & | & 1 \2 & -2 & 3 & | & 5\end{bmatrix}\]Here, the vertical bar is conventionally used to separate the coefficient matrix from the constants for clarity.

Key concepts

System of Linear Equations

A system of linear equations is essentially a collection of one or more linear equations that involve the same set of variables. In the exercise, we have three equations representing a system where the variables are \(x\), \(y\), and \(z\). Each equation expresses a relationship among these variables:

• The first equation: \(3x + 4y + 5z = 7\)
• The second equation: \(-x + y - 3z = 1\)
• The third equation: \(2x - 2y + 3z = 5\)

The goal is to find values for \(x\), \(y\), and \(z\) that satisfy all equations simultaneously. Such systems can be visualized as lines, planes, or hyperplanes in multidimensional space, with their solution representing the intersection point or region.

Coefficients

Coefficients are the numbers that multiply the variables in an equation. They play a critical role in defining the slope and direction of lines or planes represented by linear equations. In our given system:

• The coefficients for the first equation \(3x + 4y + 5z = 7\) are 3, 4, and 5, associated with \(x\), \(y\), and \(z\) respectively.
• The second equation \(-x + y - 3z = 1\) has coefficients of -1, 1, and -3.
• The third equation \(2x - 2y + 3z = 5\) involves coefficients of 2, -2, and 3.

These coefficients form the backbone of the matrix representation we will later discuss, and each row of the matrix will correspond directly to the coefficients of these equations.

Constants

In a system of linear equations, the constants are the terms independent of variables, appearing on the opposite side of the equation. These determine where the lines or planes intersect on the axes, essentially shifting the location of each plane. For our system:

• The constant in the first equation is 7.
• The second equation has a constant of 1.
• The third equation’s constant is 5.

During the conversion to an augmented matrix, these constants will appear in a separate column, helping to define the system's solutions. They are critical in aligning each plane/line in the coordinate system.

Matrix Representation

Matrix representation is a method of expressing a system of linear equations in a structured manner, which can be advantageous for solving, especially with computer algorithms or analytical techniques. It organizes the coefficients and constants into a neat grid.In our exercise, converting the system into an augmented matrix involves:

• Writing each equation's coefficients in a row, in the same order of variables: \(x, y, z\).
• Placing the constant term in a column separated by a vertical bar, making it easy to distinguish from the coefficients.

The resulting matrix representation of the given system is:\[\begin{bmatrix}3 & 4 & 5 & | & 7 \-1 & 1 & -3 & | & 1 \2 & -2 & 3 & | & 5\end{bmatrix}\]This augmented matrix effectively captures all the information required to solve the system, much like how the original equations do, but in a compact and computationally friendly format.