Problem 33 - College Algebra

Question

Solve the system by Gaussian elimination. \(\left[\begin{array}{lll|l}1 & 0 & 0 & 31 \\ 0 & 1 & 1 & 45 \\ 0 & 0 & 1 & 87\end{array}\right]\)

Step-by-Step Solution

Answer

The solution is \(x = 31\), \(y = -42\), \(z = 87\).

Step 1: Identify matrix format

The matrix given is an augmented matrix derived from a system of linear equations. Each row represents an equation and the last column represents the constant terms. The given matrix represents:1. \(x = 31\)2. \(y + z = 45\)3. \(z = 87\)

Step 2: Solve for individual variables

From row 1, we have already the value for \(x\):\[x = 31\]From row 3, we can directly extract the value of \(z\):\[z = 87\]Plugging the value of \(z\) into row 2: \(y + 87 = 45\), we solve for \(y\):\[y = 45 - 87 = -42\]

Step 3: Review the solution

After computing all variables, the solution to the system of equations is:\(x = 31\), \(y = -42\), and \(z = 87\). Verify by substituting back into the original equations to ensure correctness.

Key Concepts

Augmented Matrix

In mathematics, an augmented matrix is a useful tool to solve systems of linear equations. It's an extended matrix that represents a set of linear equations. Imagine you have a matrix with initial columns representing the coefficients of variables and an additional column added for the constant terms. This extra column contains the solutions for each equation, making it easy to see everything at a glance.
Think of it this way: each row in the augmented matrix corresponds to one specific linear equation, and each column stands for a different variable, except the last one which holds the constant term. For example, if you have a system of three equations with three variables, your matrix will have four columns in total, with the fourth column dedicated to the constants.

Here's how it helps:

• Simplifies the representation of linear systems
• Facilitates the use of methods like Gaussian elimination
• Provides a clear visual organization of equations and constants

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. These systems are fundamental in mathematics and appear frequently in various fields, like physics, engineering, and economics. Each equation can be imagined as a line (or plane, depending on the number of variables), and the solution to the system is where these lines (or planes) intersect.
For instance, if you have two variables, each equation represents a line on a graph. The solution is a point where both lines meet. For three variables, each equation can be visualized as a plane, and the solution is where these planes intersect in space.

Key characteristics of systems of linear equations:

• Can have a single solution, infinitely many solutions, or no solution
• The goal is to find values for each variable that satisfy all equations simultaneously
• Often solved using methods like substitution, elimination, or matrix techniques like Gaussian elimination

Matrix Format

The matrix format is an efficient way of organizing and managing equations, especially when dealing with complex systems. In this format, each row represents a separate linear equation, and each column corresponds to the coefficients of each variable in those equations. By writing our equations in this neat, tabular format, it becomes clearer and simpler to apply mathematical methods like Gaussian elimination.
Consider a scenario where you have a system of three equations with three variables. In matrix format, your equations become rows, making it convenient to perform operations directly on the rows to simplify to a solution.

Benefits of using matrix format:

• Streamlines calculations by keeping the equations aligned and organized
• Enables the application of linear algebra methods that deal with larger datasets efficiently
• Useful for computerized solving via algorithms