Problem 37 - 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. $$\begin{aligned} &2 x-3 y=2\\\ &4 x-6 y=1 \end{aligned}$$

Step-by-Step Solution

Answer

The system has no solution because it is inconsistent.

Step 1: Write the Augmented Matrix

First, we convert the given system of equations into an augmented matrix. The system \( \begin{aligned} &2x - 3y = 2 \ &4x - 6y = 1 \end{aligned} \), converts to: \[ \left[ \begin{array}{cc|c} 2 & -3 & 2 \ 4 & -6 & 1 \end{array} \right] \].

Step 2: Row Operation to Simplify

To eliminate the variable \(x\) from the second row, perform the row operation \( R_2 \leftarrow R_2 - 2R_1 \). This results in: \[ \left[ \begin{array}{cc|c} 2 & -3 & 2 \ 0 & 0 & -3 \end{array} \right] \].

Step 3: Check for Inconsistency

Notice that the second row \([0\ 0|-3]\) is inconsistent because it implies \(0 = -3\), which is contradictory. This indicates that the system has no solution.

Key Concepts

Row Operations

Row operations are a crucial tool in solving systems of equations using augmented matrices. They include operations such as swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row from another row. Row operations help in simplifying a matrix, making it easier to identify solutions of a system or to recognize certain characteristics like inconsistency or dependency in equations.

In the provided exercise, the row operation used was:

• Subtract twice the first row from the second row (i.e., \( R_2 \leftarrow R_2 - 2R_1 \)).

This operation targets elimination of variable terms to help isolate variables or to highlight contradictions within the equations. Row operations maintain the equivalence and balance of the system while allowing for a more manageable form.

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. Solving a system of equations means finding values for the unknowns that satisfy all equations simultaneously.

For the system:

• \( 2x - 3y = 2 \)
• \( 4x - 6y = 1 \)

We are given in this exercise, a developed approach to solving it involves transforming it into an augmented matrix and applying row operations to simplify it, aiming toward a potential solution. Often, you might find a solution set, a unique solution, or recognize infinite solutions through this process.

No Solution

A system of equations can have different types of solutions: a unique solution, infinitely many solutions, or no solution at all. When a system has no solution, it is termed as 'inconsistent'. This means there are no possible values for the variables that can satisfy all equations simultaneously.

In this exercise, after performing the row operation, the augmented matrix reveals an inconsistency in the form of the row \([0 ext{ }0|-3]\). This implies a contradiction: \(0 = -3\), which is not possible. This contradiction clearly shows that there is no possible solution for this system of equations. Recognizing no solution early in the problem helps save time and accurately depict the nature of the relationship between the given equations.