Problem 39 - 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. $$\begin{array}{c} 2 x-y=1 \\ -4 x+2 y=-2 \end{array}$$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Write the Augmented Matrix

The given system of equations is:\\(2x - y = 1\) \\(-4x + 2y = -2\)\Write the augmented matrix for this system as follows:\[\begin{bmatrix}2 & -1 & | & 1 \-4 & 2 & | & -2 \\end{bmatrix}\]

Step 2: Perform Row 1 Operations

Our first step will be to simplify Row 1 for easier manipulation. Currently:\(R1 = [2, -1, |, 1]\). Divide Row 1 by 2 to simplify it:\[R1 \rightarrow \frac{1}{2}R1 \\begin{bmatrix}1 & -0.5 & | & 0.5 \-4 & 2 & | & -2 \\end{bmatrix}\]

Step 3: Eliminate x from Row 2

Now, eliminate the x term in Row 2 using Row 1.\Add \(4\times R1\) to \(R2\):\\[R2 \rightarrow R2 + 4 \times R1 \\begin{bmatrix}1 & -0.5 & | & 0.5 \0 & 0 & | & 0 \\end{bmatrix}\]

Step 4: Interpret and Solve the System

The matrix \(\begin{bmatrix}1 & -0.5 & | & 0.5 \ 0 & 0 & | & 0\end{bmatrix}\) implies that the system has infinitely many solutions. This happens when the rows are multiples of each other, indicating linear dependence of equations.

Key Concepts

Augmented Matrix

An augmented matrix is a compact and organized way to represent a system of linear equations. It consists of coefficients of the variables and constants from the equations, usually in a rectangular array format. For the system in question, \(2x - y = 1\) and \(-4x + 2y = -2\), the augmented matrix is represented as:\[\begin{bmatrix}2 & -1 & | & 1 \ -4 & 2 & | & -2 \end{bmatrix}\]

• The vertical line in the matrix separates the coefficients of the variables on the left from the constants on the right.
• This form is particularly useful for applying row operations and reducing the matrix to solve the system.

Row operations can then be applied to transform this matrix into a simpler form which makes it easier to find solutions to the system.

System of Equations

A system of equations consists of multiple equations involving the same set of variables. Solving such a system means finding a set of values for the variables that satisfy all of the equations involved simultaneously. In this problem, the system is:

• \(2x - y = 1\)
• \(-4x + 2y = -2\)

To solve this system using the augmented matrix approach, we use row operations to simplify the system, typically aiming for a row-echelon form or reduced row-echelon form.
In row-echelon form, the goal is to create zeros below each pivot position (leading non-zero entry) in the matrix. Once this is achieved, the solutions to the equations are much more easily identified.

Infinite Solutions

When a system of equations has infinite solutions, it means there are countless sets of variable values that satisfy all equations in the system. This typically happens when the equations describe the same geometric object, like a line, plane, etc. In our system:

• The second equation is a multiple of the first, indicating both describe the same line in a 2D space.
• After performing row operations, we end up with a row of zeros, suggesting these overlapping solutions.

The row of zeros in the reduced matrix \(\begin{bmatrix}1 & -0.5 & | & 0.5 \ 0 & 0 & | & 0\end{bmatrix}\) implies a dependency that allows multiple solutions to fit.

Linear Dependence

Linear dependence occurs when one of the rows (or equations) in a system can be expressed as a linear combination of the others. For this exercise, the equation \(-4x + 2y = -2\) is just \(-2\) times the first equation \(2x - y = 1\).

• This dependency reveals that the equations do not provide independent directions, resulting in a lack of a unique solution.
• Instead, it demonstrates that both equations represent the same line, leading to the infinite number of solutions we observed.

Linear dependence is a key concept in understanding why certain systems of equations don't have unique solutions and must be solved through alternative means, such as recognizing the repetitive pattern.