Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{rr} w+2 x-3 y+z= & 18 \\ 3 w \quad-5 y=8 \\ w+x+y+2 z= & 15 \\ -w-x+2 y+z= & -3 \end{array}\right. $$

Step-by-Step Solution

Answer

The augmented matrix for the system of equations is:\n \[\begin{align*}1 & 2 & -3 & 1 & 18 \3 & 0 & -5 & 0 & 8 \1 & 1 & 1 & 2 & 15 \-1 & -1 & 2 & 1 & -3 \\end{align*}\]

Step 1: Write first row

The first row is obtained from the coefficients of the first equation, which are 1 for w, 2 for x, -3 for y, 1 for z and 18 for the constant.

Step 2: Write second row

The second row is obtained from the coefficients of the second equation. Since the x and z terms are missing in this equation, their coefficients are 0. So, the coefficients are 3 for w, 0 for x, -5 for y, 0 for z and 8 for the constant.

Step 3: Write third row

The third row is obtained from the coefficients of the third equation, which are 1 for w, 1 for x, 1 for y, 2 for z and 15 for the constant.

Step 4: Write fourth row

The fourth row is obtained from the coefficients of the fourth equation. Here, the w, x, and y terms have negative coefficients, they are -1 for w, -1 for x, 2 for y, 1 for z and -3 for the constant.

Key Concepts

Linear Equations

When we talk about linear equations, we refer to algebraic expressions that represent straight lines when graphed on a coordinate plane. Each equation involves variables and constants and follows the standard form of \(Ax + By + Cz = D\), where \(A\), \(B\), and \(C\) are the coefficients of the variables \(x\), \(y\), and \(z\) respectively, and \(D\) is the constant term. The beauty of linear equations lies in their simplicity; they describe a relationship in which each variable is raised only to the first power and the graph of such an equation is always a straight line.

One critical aspect of understanding linear equations is learning how to manipulate and solve them. The initial step involves identifying the coefficients and the constant. For instance, in the first equation of our example \(w + 2x - 3y + z = 18\), '1' is the coefficient of \(w\), '2' for \(x\), '-3' for \(y\), '1' for \(z\), and '18' is the constant. These equations often represent real-world problems, such as predicting expenses, calculating distances, or even estimating population growth.

System of Equations

A system of equations is a set of two or more equations that have a common solution. In other words, we are looking for a set of values that satisfy all equations in the system simultaneously. With a system of linear equations, such as the one we have in our exercise, the goal is to find the values of \(w\), \(x\), \(y\), and \(z\) that solve all four equations together.

To do this, various methods can be employed, ranging from graphing and substitution to elimination and matrix operations. The choice of method often depends on the complexity of the system and the number of variables involved. For example, while graphing might work for systems with two variables, it becomes impractical with three or more variables, where algebraic methods or matrix solutions are more efficient. Systems of equations have vast applications in fields such as economics, engineering, and physics, where relationships between multiple factors need to be assessed and quantified.

Matrix Representation

Matrix representation offers a powerful way to handle systems of linear equations, converting them into a compact and manipulable form. A matrix is simply an array of numbers arranged in rows and columns. In the context of linear systems, the augmented matrix is frequently used; it includes the coefficients of the variables along with the constant terms from the system's equations.

To illustrate this, let's revisit the exercise. Each row of the augmented matrix corresponds to an equation of the system, where the columns represent the coefficients of \(w\), \(x\), \(y\), and \(z\), and the last column represents the constants. The augmented matrix for our system looks like this:\[\begin{array}{cccc|c}1 & 2 & -3 & 1 & 18 \3 & 0 & -5 & 0 & 8 \1 & 1 & 1 & 2 & 15 \-1 & -1 & 2 & 1 & -3 \end{array}\]The advantage of this representation is that we can use matrix operations to solve the system, such as row reduction to echelon form or reduced echelon form, or even matrix inversion in some cases. Matrix representation simplifies complicated systems and is especially useful when dealing with higher-dimensional systems that are not easily visualized.