Problem 24 - Precalculus with Limits

Question

In Exercises \(21-26,\) write the system of linear equations represented by the augmented matrix. (Use variables \(x, y, z\) and \(w,\) if applicable.) $$\left[\begin{array}{rrrrr}{4} & {-5} & {-1} & {\vdots} & {18} \\ {-11} & {0} & {6} & {\vdots} & {25} \\ {3} & {8} & {0} & {\vdots} & {-29}\end{array}\right]$$

Step-by-Step Solution

Answer

The system of linear equations represented by the given augmented matrix is: \n1) \(4x - 5y - z = 18\)\n2) \(-11x + 6z = 25\)\n3) \(3x + 8y = -29\)

Step 1: Understanding the Augmented Matrix

An augmented matrix represents a system of linear equations. Each row corresponds to a separate equation in the system, and each column corresponds to a particular variable. The number \(4, -5, -1\) on the first row represent the coefficients of the variables \(x, y, z\) respectively for the first equation and \(18\) is the constant term. Similarly in successive rows; the first three numbers represent the coefficients and the last number represent the constant term.

Step 2: Writing Out the Equations

Translate the augmented matrix into system of linear equations. The rows become equations, like this: The first row \([4,-5,-1, 18]\) corresponds to the equation \(4x - 5y - z = 18\). The second row \([-11, 0, 6, 25]\) corresponds to the equation \(-11x + 0y + 6z = 25\) (which simplifies to \(-11x + 6z = 25\)). The third row \([3, 8, 0, -29]\) corresponds to the equation \(3x + 8y + 0z = -29\) (which simplifies to \(3x + 8y = -29\)).

Key Concepts

System of Linear Equations

Understanding how a system of linear equations works is crucial for solving a variety of mathematical and real-world problems. At its core, a system of linear equations is a collection of two or more linear equations involving the same set of variables. Each equation represents a straight line if graphed in two dimensions, and the goal is often to find where these lines intersect, which represents the solution to the system.

In a practical context, each equation could represent constraints or conditions that need to be satisfied simultaneously. For instance, if the variables represent quantities of products and the equations represent different limitations like cost, materials, or labor, solving the system can help in optimizing a production process.

To solve these systems, there are several methods available like graphing, substitution, elimination, and matrix operations. Each has its own applications and utility based on the complexity of the system.

Matrix Representation

Converting a system of linear equations into matrix form is a powerful way to simplify and solve complex problems. In this matrix representation, the coefficients of the variables are laid out in a grid format, with the constants that are normally on the right side of each equation collected in a column on the far right; this is the augmented column.

Matrices can be manipulated through operations such as row reduction, which can make it easier to identify the solutions to the system. The primary benefit of using matrix representation is that it can handle systems with many equations and variables in a structured manner, making it an essential tool in fields such as engineering, computer science, and economics.

Algebra

Algebra plays a fundamental role in the study of mathematics and more so when dealing with systems of linear equations and matrices. It provides the symbolic language and operations needed to express and solve mathematical problems. From combining like terms to applying the distributive property, the principles of algebra are used to manipulate equations into a more solvable form.

Algebra is instrumental when performing operations on an augmented matrix, such as scaling rows to create leading ones, or adding and subtracting rows to eliminate variables, ultimately aiming to find the values of unknowns that satisfy all equations simultaneously.

Variables

Variables are the symbols that represent unknown quantities in algebraic expressions and equations. They are at the heart of algebra and serve as placeholders for values that can change or that we wish to solve for. In the context of a system of linear equations, variables represent the dimensions along which the system extends. For instance, in a two-variable system, we typically use x and y, while in three dimensions, z is added.

In the exercise provided, the variables are x, y, and z. Solving the system involves finding the specific values of these variables that make all the equations hold true simultaneously. The ability to work comfortably with variables is essential for solving any algebraic problem, including those involving matrix representations.