Problem 12 - 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. \(\left[\begin{array}{rrrr|r}4 & 1 & 5 & 1 & 6 \\ 1 & -1 & 0 & -1 & 8 \\ 3 & 0 & 0 & 7 & 4 \\ 0 & 0 & 11 & 5 & 3\end{array}\right]\)

Step-by-Step Solution

Answer

The system of linear equations that the given augmented matrix represents is: \n1. \(4x + y + 5z + w = 6\)\n2. \(x - y - w = 8\)\n3. \(3x + 7w = 4\)\n4. \(11z + 5w = 3\)

Step 1: Identify the Variables

The variables for this system of equations are \(x, y, z, w\). This is determined by the number of columns in the coefficient matrix (four in this case). The final column represents the constants on the right-hand side of the equation.

Step 2: Formulate the linear equations

Formulate the linear equations from the augmented matrix using the variables. Each row represents a separate equation in the system. Translate each element of the rows into their respective coefficients. The equations will be as follows: \n1. \(4x + y + 5z + w = 6\)\n2. \(x - y - w = 8\)\n3. \(3x + 7w = 4\)\n4. \(11z + 5w = 3\)

Step 3: Recap the System of Linear Equations

The system of linear equations that the given augmented matrix represents is: \n1. \(4x + y + 5z + w = 6\)\n2. \(x - y - w = 8\)\n3. \(3x + 7w = 4\)\n4. \(11z + 5w = 3\)

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool used to simplify and solve systems of linear equations. It combines the coefficients of variables and the constants on the right-hand side into one convenient display.
Imagine it like this:

• We organize the coefficients of each variable in the system into rows and columns.
• The rightmost column contains the constants from every equation.

This format makes it simpler to perform operations to solve the system. For instance, the augmented matrix given in the exercise: \[\begin{bmatrix}4 & 1 & 5 & 1 & \vert & 6 \1 & -1 & 0 & -1 & \vert & 8 \3 & 0 & 0 & 7 & \vert & 4 \0 & 0 & 11 & 5 & \vert & 3\end{bmatrix}\]represents four linear equations. The vertical bar separates the coefficients from the constants, making it clear which numbers correspond to the variables and which to the constants.
This setup is especially useful when applying matrix operations like row reduction to find solutions.

Matrices in Algebra

Matrices are essential in modern algebra and mathematics. They serve multiple purposes, but in solving systems of linear equations, they are indispensable. Here’s how they function in this context:

• Each matrix represents a vector of equations, allowing for structured manipulation.
• The rows of a matrix can be viewed as individual linear equations, with their positions aiding in systematic transformation.
• Column positions reflect different variables involved in equations.

Working with matrices simplifies what can be cumbersome calculations when dealing with many equations and variables. It’s like translating a word problem into a formula – once in matrix form, we can easily perform operations to 'simplify' or 'reduce' the data, eventually leading us to solutions.
In practical applications beyond just linear equations, such as computer graphics, network theory, and data analysis, matrices provide a robust framework for managing complex information efficiently.

Solving Linear Equations

Solving a system of linear equations involves finding the values of variables that satisfy all the given equations simultaneously. A popular approach uses matrices to organize the work.
Here’s a basic rundown of how solving typically proceeds:

• Write the system of equations in augmented matrix form.
• Perform row operations (like swapping, multiplying, or adding rows) to transform the matrix into a simpler form, often called row-echelon form or reduced row-echelon form.
• Once the matrix is simplified, interpret it back into equations to find the variable values.

The beauty of this method lies in its efficiency and near-universal application. Whether dealing with three equations or a hundred, the structured approach remains largely the same. This consistency makes it a staple technique in algebra and helps in understanding the intricate relationships between the variables in a system.