Problem 5 - Algebra and Trigonometry

Question

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{array}{r} {5 x-2 y-3 z=0} \\ {x+y=5} \\ {2 x-3 z=4} \end{array}\right. $$

Step-by-Step Solution

Answer

The augmented matrix is \[ \left[ \begin{{array}}{{ccc|c}} 5 & -2 & -3 & 0 \ 1 & 1 & 0 & 5 \ 2 & 0 & -3 & 4 \end{{array}} \right] \].

Step 1: Identify Variables and Coefficients

First, identify the coefficients of the variables (x, y, z) in each equation. These coefficients will form the elements of the matrix. From the first equation, the coefficients are 5, -2 and -3 respectively for x, y, and z; for the second, they are 1, 1, and 0 (because z isn't present in the equation), and for the third, coefficients are 2, 0, and -3 (since y isn't present). Also, note the constants on the right-hand side of the equations: 0, 5, and 4 respectively.

Step 2: Construct Augmented Matrix

Now, form the augmented matrix using these coefficients and constants. Augmented matrices have vertical lines that separate the coefficients and constants. Write each row as the coefficients from each equation and the constant after the vertical line. The final augmented matrix will be:\[ \begin{{array}}{{ccc|c}}5 & -2 & -3 & 0 \1 & 1 & 0 & 5 \2 & 0 & -3 & 4\end{{array}} \]

Key Concepts

Linear Equations

Understanding linear equations is pivotal for solving a variety of problems in mathematics and real-world applications. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. Linear equations can be recognised by the fact that the variable has an exponent of one.

For example, in the system of equations given in the exercise, the equations are linear with regard to variables x, y, and z. A key feature of linear equations is that they graphically represent straight lines on a Cartesian plane, and when we have a system of such equations, we're essentially looking for points where these lines intersect — which corresponds to the common solution set for the system.

Moreover, when dealing with linear equations, we can manipulate them in many ways without changing the solution: we can add or subtract equations, multiply or divide by non-zero constants, and substitute one equation into another. These operations are fundamental when we utilize matrices to simplify and solve the equations.

Matrix Representation

The matrix representation of a system of linear equations allows us to condense the information contained in the system into a compact form that is particularly useful for computational methods. Each row in a matrix corresponds to an equation in the system, while each column represents the coefficients of a particular variable across those equations.

Creating a Matrix from Equations

In essence, to create a matrix from a system of equations, you list down the coefficients of the variables in the same order for each equation, treating any missing variable as having a coefficient of zero. So for the second equation in our exercise, because it lacks a z term, we insert a zero in the position that represents the coefficient of z.

This structure allows us to apply matrix operations to solve the equations, which can be simpler and more systematic than other methods. Especially with larger systems, matrix methods like Gaussian elimination, matrix inverses, or determinants become indispensable tools.

Systems of Equations

A system of equations is a set of two or more equations that share a common set of unknowns and thus are solved together. These systems can have one solution, no solution, or infinitely many solutions.

Systems of equations can be visually analyzed in the case of two or three variables. The solution (if one exists) can be seen as the intersection point of lines (in two dimensions) or planes (in three dimensions). When we cannot easily visualize the system, as in cases with more than three variables, we turn to algebraic methods, such as substitution, elimination, and as seen in the exercise, matrix representations which lead us to the augmented matrix.

Creating an augmented matrix is a crucial first step in using matrix operations to solve these systems. The vertical line in the augmented matrix is crucial as it separates the coefficient section from the solution constants, setting the stage for methods such as row reduction to find the solution(s) to the system.