Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{r} x+2 y=1 \\ 2 x+4 y=3 \end{array}$$

Step-by-Step Solution

Answer

The system has no solution; it's inconsistent.

Step 1: Write the Augmented Matrix

First, represent the given system of equations in an augmented matrix form. The system is: \[ x + 2y = 1 \] \[ 2x + 4y = 3 \] So the augmented matrix is: \[ \begin{bmatrix} 1 & 2 & | & 1 \ 2 & 4 & | & 3 \end{bmatrix} \]

Step 2: Apply Row Operations to Achieve Row Echelon Form

We want the leading coefficient of each row to be 1. We start by modifying the second row: replace it with the result of subtracting 2 times the first row from the second row. - From row 2, subtract 2 \( \times \) row 1: \[ \begin{bmatrix} 1 & 2 & | & 1 \ 2 - 2(1) & 4 - 2(2) & | & 3 - 2(1) \end{bmatrix} \] Results in: \[ \begin{bmatrix} 1 & 2 & | & 1 \ 0 & 0 & | & 1 \end{bmatrix} \]

Step 3: Identify Inconsistency

Examine the resulting matrix. Notice that the second row translates to the equation \(0 = 1\), indicating an inconsistency. This implies that the system of equations has no solution.

Key Concepts

Augmented Matrix

An augmented matrix is a crucial tool in solving systems of equations. It allows us to represent the system in a compact form using rows and columns. Each equation in the system is transcribed into a row of the matrix. In an augmented matrix, there is an extra column that represents the constants on the right-hand side of the equations. This additional part helps differentiate it from a regular matrix, and sets it up nicely for row operations.
For example, in the system given:

• Equation 1: \( x + 2y = 1 \)
• Equation 2: \( 2x + 4y = 3 \)

We express these equations as an augmented matrix where each row corresponds to an equation, and each column represents the coefficients of the variables. The matrix becomes:\[\begin{bmatrix} 1 & 2 & | & 1 \2 & 4 & | & 3 \end{bmatrix}\]The vertical line helps indicate the separation between the coefficients of variables and the constants, enhancing clarity in operations.

System of Equations

A system of equations is formed when two or more mathematical equations share the same set of variables. The goal is usually to find the values of these variables that satisfy all the equations simultaneously. Systems can range from simple to very complex based on the number of equations and variables involved.
Consider our example, where the system consists of:

• \( x + 2y = 1 \)
• \( 2x + 4y = 3 \)

This system of linear equations represents two lines when graphed on a coordinate plane. Solving the system usually involves finding the intersection point of the lines, indicating the values of \(x\) and \(y\) that satisfy both equations.
However, if the lines are parallel and never intersect, the system will have no solution, which is called an inconsistent system.

Inconsistent System

An inconsistent system occurs when the system of equations does not have common solutions. This implies that there is no set of variables that will satisfy all equations at once. Often, after performing row operations, this is indicated by a row in the augmented matrix that translates into an impossible statement.
In the provided example, row operations led us to the following matrix form:\[\begin{bmatrix} 1 & 2 & | & 1 \0 & 0 & | & 1 \end{bmatrix}\]
The second row suggests \(0 = 1\), a clear inconsistency. Therefore, no solutions exist where both original equations are satisfied. This kind of system, where an equation implied by the matrix is false, is what we call an inconsistent system.

Row Echelon Form

The row echelon form (REF) of a matrix is a special state that aids the solution of systems of equations greatly by simplifying the process of back-substitution. To transform a matrix into REF, follow these conditions:

• Every leading entry of a row is to the right of the leading entry of the previous row.
• All entries below a leading entry are zero.
• Rows consisting entirely of zeros, if any, are at the bottom of the matrix.

In the example, we applied row operations to reach the REF:\[\begin{bmatrix} 1 & 2 & | & 1 \0 & 0 & | & 1 \end{bmatrix}\]
The row echelon form helped us identify that the system is inconsistent by highlighting the false statement. Achieving this form makes it easier to identify solutions or the nature of the system, as each row provides a straightforward interpretation of the equations.