Problem 22 - A Graphical Approach to College Algebra

Question

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rr|r} 1 & -1 & 2 \\ 0 & 1 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The solution is \( x = 2 \) and \( y = 0 \).

Step 1: Identify the System from the Augmented Matrix

The given augmented matrix represents the following system of equations: \( x - y = 2 \) and \( y = 0 \).

Step 2: Solve for y

From the second equation \( y = 0 \), we find the value of \( y \). It is already isolated and ready to be substituted into the first equation.

Step 3: Substitute y and Solve for x

Substitute \( y = 0 \) into the first equation \( x - y = 2 \) to get \( x - 0 = 2 \), which simplifies to \( x = 2 \).

Step 4: Write the Solution

The solution to the system is \( x = 2 \) and \( y = 0 \).

Key Concepts

Row Echelon Form

Understanding the row echelon form is a key part of solving linear systems using matrices. In this structure, the matrix appears in a specific format that makes it easier to solve the equations it represents. A matrix is said to be in row echelon form if:

• All non-zero rows are above any rows of all zeroes.
• The leading entry (also known as the pivot) of a non-zero row is always to the right of the leading entry of the previous row.
• The leading entry in any non-zero row is 1 (this is often achieved through row operations).

In the given augmented matrix \[\begin{array}{rr|r} 1 & -1 & 2 \0 & 1 & 0 \end{array}\]we can see it is already in row echelon form, as the leading entry in both rows is 1, and the second row's leading entry (in the second column) is to the right of the first row's leading entry (which is in the first column). This structured form simplifies the process of solving the linear system using back-substitution.

Linear System

A linear system is a collection of linear equations involving the same set of variables. Linear systems can consist of any number of equations and unknowns. Solving these systems is a fundamental aspect of algebra and calculus.

In this exercise, the linear system is comprised of the equations represented by the augmented matrix:\[\begin{array}{rr|r}1 & -1 & 2 \0 & 1 & 0\end{array}\]which translates to the following two equations:

• \(x - y = 2\)
• \(y = 0\)

By understanding these equations, you can solve for both variables, \(x\) and \(y\), using techniques such as substitution or elimination (in larger systems). Back-substitution, as shown in this exercise, helps attain the solution based on a simpler, step-by-step method.

Augmented Matrix

An augmented matrix is a powerful tool used to solve linear systems of equations. It combines the coefficients of the variables and the constants from the equations into a single matrix form. This makes it easier to manipulate and solve equations using row operations.

The augmented matrix in this exercise is:\[\begin{array}{rr|r}1 & -1 & 2 \0 & 1 & 0\end{array}\]Here, the horizontal line separates the coefficients on the left from the constants on the right. This matrix form directly corresponds to the original system of two equations.

For this matrix, the first row implies \(x - y = 2\), giving us the relationship between \(x\) and \(y\). The second row immediately tells us that \(y = 0\). By using the augmented matrix effectively, we can simplify the process of finding the values of \(x\) and \(y\), often leading to quicker and more intuitive solutions.