Problem 38 - A Graphical Approach to College Algebra

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 is inconsistent; no solution exists.

Step 1: Write the Augmented Matrix

First, express the system of equations as an augmented matrix. The given system of equations is:\[ \begin{align*} x + 2y &= 1 \ 2x + 4y &= 3 \end{align*} \]This can be written in matrix form as:\[ \begin{bmatrix} 1 & 2 & | & 1 \ 2 & 4 & | & 3 \end{bmatrix} \]

Step 2: Apply Row Operation to Create a Leading 1 in First Column

Make sure the first element of the first column is already 1. Our matrix is fine in this respect, so no changes are necessary. Proceed to the next row operation.

Step 3: Eliminate the First Column Entry of the Second Row

We need to make the element below the leading 1 in the first column zero. Subtract two times the first row from the second row:\[ R_2 \leftarrow R_2 - 2R_1 \]This operation gives us:\[ \begin{bmatrix} 1 & 2 & | & 1 \ 0 & 0 & | & 1 \end{bmatrix} \]

Step 4: Determine Consistency of the System

Look at the second row of the resulting matrix. It is of the form \(0x + 0y = 1\), which implies an inconsistency (0 cannot equal 1). Thus, there is no solution to the system of equations.

Key Concepts

Row Operations

Row operations are essential tools that help solve systems of equations using matrices. By applying transformations to the rows of a matrix, you can transition it into a simpler form. There are three types of row operations: swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row to another. These operations maintain the equality of the system and are instrumental in reaching a solution.

In our exercise, row operations were used on an augmented matrix. We initially set up the matrix based on the equations given. The primary goal was to create zeros below the leading coefficient (also known as a pivot) by manipulating rows. This involves calculating and applying the necessary operations to simplify the matrix.

When dealing with systems of equations, using row operations ensures accuracy and efficiency in finding solutions. They are foundational for methods like Gaussian elimination and Gauss-Jordan elimination.

System of Equations

A system of equations consists of two or more equations that share the same variables. The objective is to find values for these variables that satisfy all the equations in the system simultaneously. Systems can be solved using various methods, such as substitution, elimination, and matrices.

In our example, the system given was:

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

This structure forms two lines in the coordinate plane. We sought a point where these lines intersect, which would represent the solution to the system.

Using matrices to solve systems of equations is efficient, especially for larger systems. The system is rewritten as an augmented matrix, allowing for structured manipulation and solving through row operations.

Inconsistent System

An inconsistent system is one where there are no solutions. This happens when the equations contradict each other, such as the case when simplified to a form like \( 0 = 1 \), which is logically impossible.

In the exercise, the system's augmented matrix was reduced to:

• \[ \begin{bmatrix} 1 & 2 & | & 1 \ 0 & 0 & | & 1 \end{bmatrix} \]

The second row translates to \( 0x + 0y = 1 \), highlighting the inconsistency. Since no values for \( x \) or \( y \) can satisfy this equation, the original system has no solution.

Understanding inconsistent systems is crucial as it saves time and effort during solving. When recognized early, it prevents unnecessary computations, indicating clearly that no intersection point (or solution) exists for the equations involved.