Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} -5 x+8 y=3 \\ 10 x+6 y=5 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 1\) and \(y = \frac{1}{2}\).

Step 1: Write the system in matrix form

Start by writing the system of equations as an augmented matrix. The system of equations given is: \[-5x + 8y = 3\] and \[10x + 6y = 5\]. This is represented by the matrix:\[\begin{bmatrix}-5 & 8 & | & 3 \10 & 6 & | & 5 \\end{bmatrix}\]

Step 2: Perform row operations to simplify the matrix

The first goal is to create a leading 1 in the first row, first column. We can achieve this by multiplying the first row by \(-\frac{1}{5}\):\[\begin{bmatrix}1 & -\frac{8}{5} & | & -\frac{3}{5} \10 & 6 & | & 5 \\end{bmatrix}\]Next, eliminate the first element of the second row by replacing Row 2 with Row 2 - 10*Row 1:\[\begin{bmatrix}1 & -\frac{8}{5} & | & -\frac{3}{5} \0 & 22 & | & 11 \\end{bmatrix}\]

Step 3: Continue row operations to find a solution

Now, make the leading coefficient of the second row equal to 1 by dividing the entire second row by 22:\[\begin{bmatrix}1 & -\frac{8}{5} & | & -\frac{3}{5} \0 & 1 & | & \frac{1}{2} \\end{bmatrix}\]Next, simplify the first row by eliminating the \(-\frac{8}{5}\) from the \(y\) coefficient by replacing Row 1 with Row 1 + \(\frac{8}{5}\)\( * \text{{Row 2}}\):\[\begin{bmatrix}1 & 0 & | & 1 \0 & 1 & | & \frac{1}{2} \\end{bmatrix}\]

Step 4: Interpret the solution

The matrix \[\begin{bmatrix} 1 & 0 & | & 1 \0 & 1 & | & \frac{1}{2} \\end{bmatrix}\] indicates the solution to the system of equations. The variables correspond to the entries from left to right in each row, giving the solution: \(x = 1\) and \(y = \frac{1}{2}\).

Key Concepts

Matrix Representation

When solving systems of equations, transforming them into a matrix can simplify the process significantly. A matrix is essentially a structured array of numbers, arranged into rows and columns. This representation offers a compact way to handle the coefficients of variables in each equation. For example, consider the equations:

• \(-5x + 8y = 3\)
• \(10x + 6y = 5\)

These can be transformed into a matrix form. The matrix setup involves creating groups that line up the coefficients of the variables, separated by a vertical line to denote the constant on the other side of the equation. Thus, the augmented matrix for this system becomes:\[\begin{bmatrix}-5 & 8 & | & 3 \10 & 6 & | & 5\end{bmatrix}\]Here, each row represents one equation from the system, and each column corresponds to the coefficients and constants for each variable in the equations.

Row Operations

Row operations are the fundamental tools used in Gaussian elimination to simplify matrices. Essentially, these operations allow us to manipulate the rows of a matrix, helping us to solve systems of equations systematically. There are three main types of row operations:

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

In the solution to our example, we first multiplied the first row by \(-\frac{1}{5}\) to create a leading 1 at the start of the matrix:\[\begin{bmatrix}1 & -\frac{8}{5} & | & -\frac{3}{5} \10 & 6 & | & 5\end{bmatrix}\]Moving forward, we focused on eliminating coefficients beneath our leading 1 by adjusting the second row:\[\begin{bmatrix}1 & -\frac{8}{5} & | & -\frac{3}{5} \0 & 22 & | & 11\end{bmatrix}\]These operations guide the matrix towards a row-echelon form, making it straightforward to backtrack and find the solution to the system of equations.

Augmented Matrix

An augmented matrix combines the coefficients and constants of a system of equations into one matrix, separated by a vertical line. This structure allows us to handle both components of a linear system within a single matrix framework. With our example:

• The left side of the matrix holds the coefficients for the variables \(x\) and \(y\).
• The right side, after the vertical line, holds the constants from the original equations.

For our equations \(-5x + 8y = 3\) and \(10x + 6y = 5\), the augmented matrix looks like this:\[\begin{bmatrix}-5 & 8 & | & 3 \10 & 6 & | & 5\end{bmatrix}\]The vertical line in an augmented matrix is only a notational device to indicate the separation between the coefficients of variables and their respective constants. Throughout the row operations, this matrix evolves, eventually revealing the solutions for each variable in the system.

System of Equations

A system of equations consists of multiple equations that are solved together, where each equation includes one or more variables. In our particular exercise, the system was:

• \(-5x + 8y = 3\)
• \(10x + 6y = 5\)

The goal when dealing with these systems is to find out the values of \(x\) and \(y\) that satisfy both equations simultaneously. Transitioning this system into a matrix format facilitates the use of Gaussian elimination, a method dedicated to finding solutions to systems of equations.
By the end of the Gaussian elimination process, our matrix transforms fully, allowing us to read off the solutions directly. It provides a clear and systematic way to handle and solve systems with multiple equations, showing that the values \(x = 1\) and \(y = \frac{1}{2}\) satisfy both equations in our example.