Problem 11 - College Algebra

Question

In Exercises \(9-12,\) write the system of linear equations represented by the augmented matrix. Use \(x, y, z,\) and, if necessary, \(w, x, y,\) and \(z,\) for the variables. \(\left[\begin{array}{rrrr|r}1 & 1 & 4 & 1 & 3 \\ -1 & 1 & -1 & 0 & 7 \\ 2 & 0 & 0 & 5 & 11 \\ 0 & 0 & 12 & 4 & 5\end{array}\right]\)

Step-by-Step Solution

Answer

The system of linear equations represented by the augmented matrix is \(x + y + 4z + w = 3, -x + y - z = 7, 2x + 5w = 11, 12z + 4w = 5\).

Step 1: Identify the Form of the Matrix

The matrix form given is a 4x5 augmented matrix, where the last column represents the constants of the equations.

Step 2: Transform Matrix to Equations

Each row of the matrix forms an equation. The columns represent the coefficients of the variables \(x\), \(y\), \(z\), \(w\) respectively. The elements after the bar ('|') form the constants of the equations. The first row [1 1 4 1 | 3] becomes \(x + y + 4z + w = 3\). The second row [-1 1 -1 0 | 7] forms \(-x + y - z = 7\). The third row [2 0 0 5 | 11] results in \(2x + 5w = 11\). The fourth row [0 0 12 4 | 5] becomes \(12z + 4w = 5\).

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations with the same set of variables. Simply put, they consist of equations working together to find a solution that satisfies all of them at once. This system can have:

• A unique solution
• Infinitely many solutions
• No solution at all (inconsistent system)

When solving these systems, the goal is to determine the values of the variables that make all equations true simultaneously.
Imagine a linear equation as a straight line on a graph. Solving these systems means finding the point(s) where all these lines intersect. In algebra, systems of equations appear in various practical scenarios like calculating resources, understanding geometry, or solving puzzles.

Matrix to Equations Transformation

Transforming a matrix into a system of equations involves understanding that each row in the matrix corresponds to a single equation. The components of the rows are the coefficients that multiply the variables.
Let's dive into the augmented matrix, which is slightly different from a regular matrix. The augmented matrix has an extra column that consists of constants from each equation, separated by a vertical line ('|').
By translating each row:

• The first number is the coefficient of the first variable, usually denoted as \(x\).
• The second number is for \(y\), the third for \(z\), and the fourth for \(w\).
• After the '|', the number represents the constant term of that equation.

So, by reading across each row and using the right coefficients for the variables, the matrix becomes a system of equations.

Coefficients of Variables

In linear equations, coefficients are the numbers that are multiplied by the variables. They play a crucial role in forming equations out of matrices because they define how each variable contributes to the overall equation.
The matrix acts as a blueprint:

• Each entry before the '|' represents a coefficient of a corresponding variable.
• Identifying these coefficients correctly ensures the equation reflects the intended relationship among variables.
• For example, in the first row of the matrix \([1, 1, 4, 1 | 3]\), 1 is the coefficient of \(x\), 1 for \(y\), 4 for \(z\), and 1 for \(w\).

Understanding coefficients is key to translating the numeric array into meaningful algebraic expressions.

Linear Algebra Concepts

Linear algebra is a branch of mathematics dealing with vectors, matrices, and linear transformations. It provides powerful tools to work with linear systems.
Concepts include:

• Vectors: Objects representing quantities with both direction and magnitude.
• Matrices: Rectangular arrangements of numbers or symbols representing a set of linear equations.
• Linear transformations: Functions that map vectors to vectors while preserving structure.

These concepts are fundamental as they form the building blocks of matrices. By utilizing the matrix, solutions to systems of equations can often be found more efficiently.
Linear algebra helps solve not only small, manageable systems but also extremely large systems that can arise in fields like computer science, engineering, or physics. It opens up methods to represent and solve real-world problems through mathematical expressions and computations.