Problem 40 - 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. $$\begin{array}{c} x-y=1 \\ -x+y=-1 \end{array}$$

Step-by-Step Solution

Answer

Infinitely many solutions: \((x, y) = (y + 1, y)\).

Step 1: Write the Augmented Matrix

The given system of equations is \(x - y = 1\) and \(-x + y = -1\). We start by writing the system as an augmented matrix:\[\begin{bmatrix} 1 & -1 & | & 1 \-1 & 1 & | & -1 \end{bmatrix}\]

Step 2: Eliminate the First Column Below the Diagonal

We add the first row to the second row to eliminate the \(-1\) below the leading \(1\) in the first column:\[R_2 = R_2 + R_1: \begin{bmatrix} 1 & -1 & | & 1 \0 & 0 & | & 0 \end{bmatrix}\]

Step 3: Interpret the Result

The resulting matrix is:\[\begin{bmatrix} 1 & -1 & | & 1 \0 & 0 & | & 0 \end{bmatrix}\]The first row corresponds to the equation \(x - y = 1\), and the second row \(0 = 0\) is true for all \(x\) and \(y\). This indicates that there are infinitely many solutions.

Step 4: Solve for One Variable

From the equation \(x - y = 1\), we can express one variable in terms of the other, such as: \[x = y + 1\]

Step 5: Describe the Solution Set

Since \(x = y + 1\), the solution set includes all points \((x, y)\) where \(x\) is one more than \(y\).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In this exercise, we have a very simple system consisting of two linear equations:

• \( x - y = 1 \)
• \( -x + y = -1 \)

These equations are defined in terms of two variables, \(x\) and \(y\). The objective is to find values for \(x\) and \(y\) that make both equations true at the same time. In simpler cases like this one, the equations are both straight lines, and finding the solution involves looking for points where these lines intersect, if they do at all. This type of system helps us understand relationships between quantities and is fundamental in areas of algebra and calculus. By solving such a system, we are working towards determining the common values for \(x\) and \(y\) that satisfy both equations simultaneously.

Row Operations

When using matrices to solve a system of equations, row operations are crucial. These operations help us simplify the augmented matrix to reach a clearer form, often referred to as row-echelon form or reduced row-echelon form. Row operations consist of:

• Swapping two rows
• Multiplying a row by a nonzero scalar
• Adding or subtracting the multiple of one row to another row

In our exercise, we employ row operations to simplify the matrix and achieve a straightforward solution for the system. For instance, in the step-by-step solution:- Adding the first row to the second (\(R_2 = R_2 + R_1\)) results in the matrix: \[\begin{bmatrix} 1 & -1 & | & 1 \0 & 0 & | & 0 \end{bmatrix}\]These operations maintain the integrity of the system while reshaping it for easier interpretation, ultimately guiding us toward the solution. Understanding row operations is fundamental in solving systems of equations using matrices, as they are effective manipulation tools that help achieve the desired results.

Infinitely Many Solutions

The concept of having 'infinitely many solutions' arises when a system of equations turns out to not have a unique solution, nor is it inconsistent (having no solution at all). Instead, the equations describe the same relationship in some way, leading to an unlimited set of solutions. In our original exercise, the matrix simplifies to:\[\begin{bmatrix} 1 & -1 & | & 1 \0 & 0 & | & 0 \end{bmatrix}\]Here, the second row corresponds to \(0 = 0\), which is always true, regardless of the values of \(x\) and \(y\). The first row remains as \(x - y = 1\), providing a relationship that must hold. Therefore, instead of a single solution, the solutions form a line where every point satisfies \(x = y + 1\). This indicates that there are infinitely many pairings of \((x, y)\) that satisfy the system, illustrating how systems can sometimes lead to a broad range of acceptable solutions, rather than a unique one.