Problem 16 - College Algebra

Question

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{ll|l}{1} & {0} & {3} \\ {0} & {0} & {0}\end{array}\right] $$

Step-by-Step Solution

Answer

The system solution is \(x = 3\) and \(y\) is any real number.

Step 1: Analyze the augmented matrix

Given the augmented matrix \[\left[\begin{array}{ll|l}{1} & {0} & {3} \ {0} & {0} & {0}\end{array}\right]\].This matrix corresponds to a system of 2 linear equations in 2 variables, but one of the equations is a zero row, which doesn't add any information.

Step 2: Interpret the system of equations

The matrix represents the system:\[\begin{align*}1x + 0y &= 3, \0x + 0y &= 0.\end{align*}\]This simplifies to:\[\begin{align*}x &= 3.\end{align*}\]The second equation \(0 = 0\) is always true, so it doesn't affect the solution.

Step 3: Determine the solution set

Since the equation \(x = 3\) is the only non-trivial row, the variable \(y\) is a free variable. This means \(y\) can take any real value while \(x\) remains 3.

Key Concepts

Augmented Matrix

An augmented matrix is a compact and convenient way to represent a system of linear equations. The term "augmented" refers to the addition of an extra column that includes the constants from the equations' right-hand sides beside the coefficient matrix. This matrix form allows for easy application of Gaussian elimination techniques to find the solutions of the system.

To convert a system of linear equations into an augmented matrix, follow these steps:

• Write down the coefficients of each variable in a row corresponding to each equation.
• Add a vertical line to separate the coefficient matrix from the constant terms.
• Place the constants on the right side of this line.

In the exercise, the augmented matrix given represents the system \(1x + 0y = 3\) and \(0x + 0y = 0\). The elimination method involves manipulating this matrix to systematically reduce it in pursuit of the simplest form, from which solutions can be more easily deduced.

System of Linear Equations

A system of linear equations is a collection of two or more equations with the same set of unknowns. Solving these systems involves finding the values of the variables that satisfy all equations at the same time.

Each equation may represent a line, a plane, or a hyperplane in a multi-dimensional space. A system can have:

• a unique solution when lines intersect at a single point,
• infinitely many solutions when lines overlap exactly,
• no solutions when lines are parallel and distinct.

In our exercise, the system consists of two equations:
- \(x = 3\)- \(0 = 0\)
The second equation doesn't change anything, leading us to the fact that the system describes a line in a plane where the y-coordinate is free to vary, while the x-coordinate remains constant at 3.

Free Variable

A free variable in a system of equations is a variable that can take on any value; it is not constrained by the other equations in the system. During the process of solving, when a column in an augmented matrix doesn't lead to a pivot (a leading entry in a row), the corresponding variable is considered free.

In our example, we have the simplified equation \(x = 3\), while the free variable is \(y\). This means:

• \(x\) has a fixed solution: \(x = 3\),
• \(y\) can be any real number.

This flexibility is expressed mathematically by saying the system has a solution of the form \( (x, y) = (3, y) \) where \(y\) can be anything, yielding an infinite number of solutions. Free variables reflect the freedom present in systems with infinitely many solutions and play a vital role in understanding the geometry of solution sets.