Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} 2 x-y=2 \\ 3 x+2 y=17 \end{array} $$

Step-by-Step Solution

Answer

The solution is \( x = 3 \) and \( y = 4 \).

Step 1: Write the System as an Augmented Matrix

First, we take the coefficients of the equations and form the augmented matrix, which includes the constants from the right side of each equation:\[\begin{bmatrix} 2 & -1 & | & 2 \3 & 2 & | & 17 \end{bmatrix}\]

Step 2: Perform Row Operations to Eliminate y from the First Column

To eliminate the variable \( y \) from the first row, we can perform a row operation. Let's multiply the first row by 2 and add it to the second row: First Row: \( 2R_1: 4x - 2y = 4 \) New Second Row: \( (3x + 2y) + (4x - 2y) = 17 + 4 \) So, the second row becomes:\( 7x = 21 \)Now our matrix looks like this:\[\begin{bmatrix}2 & -1 & | & 2 \7 & 0 & | & 21 \end{bmatrix}\]

Step 3: Solve for x from the Second Equation

The second row of the matrix, \( 7x = 21 \), is now simple to solve:\[ x = \frac{21}{7} = 3 \]

Step 4: Substitute x into the First Equation

Substitute \( x = 3 \) into the first equation from the matrix:\( 2x - y = 2 \)\( 2(3) - y = 2 \)\( 6 - y = 2 \)\( y = 6 - 2 = 4 \)

Step 5: State the Solution to the System

The solution of the system is \( x = 3 \) and \( y = 4 \). Thus, the point of intersection of the lines represented by these equations is \((3, 4)\).

Key Concepts

Augmented Matrix

An augmented matrix is a compact way of representing a system of linear equations. It consists of the coefficients of the variables and the constants from each equation. This matrix format helps us easily apply operations to simplify and solve the system.

For example, consider the system of equations from the exercise:

• Equation 1: 2x - y = 2
• Equation 2: 3x + 2y = 17

These equations can be written as an augmented matrix:
\[\begin{bmatrix} 2 & -1 & | & 2 \ 3 & 2 & | & 17 \end{bmatrix}\]In this matrix, the vertical line separates the coefficients on the left from the constants on the right. With this representation, we can easily apply row operations, which is crucial for solving systems via Gaussian elimination.

Row Operations

Row operations are the key transformations applied to augmented matrices to simplify and eventually solve them. There are three main types of row operations:

• Swap two rows
• Multiply a row by a non-zero scalar
• Add or subtract a multiple of one row from another row

These operations help in forming a matrix in a row-echelon form or a reduced row-echelon form.

In our example, to eliminate the variable \( y \) from the first row, we multiplied the first row by 2 and added it to the second row. As a result, we obtained a simpler equation for \( x \):

First we scale the first row:\[2R_1 = 4x - 2y = 4\]Add this to the second row:\[3x + 2y + 4x - 2y = 17 + 4 = 21\]Result in the second row as:\[7x = 21\]
Now solving for \( x \) simplifies the second equation, making it easier to substitute back and find \( y \).

System of Equations

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

In our example, we have a system of two linear equations:

• 2x - y = 2
• 3x + 2y = 17

The goal is to find values for \( x \) and \( y \) that make both equations true at the same time. Gaussian elimination allows us to systematically eliminate variables through operations, making it easier to solve for the remaining variables.

This particular system is consistent and independent, meaning it has a unique solution which we found to be \( x = 3 \) and \( y = 4 \) after applying the described steps.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. It provides the fundamental tools necessary for analyzing and solving systems of linear equations.

Gaussian elimination, employed in the solution, is a classical approach rooted in linear algebra to simplify a system to a point where the solutions can be easily determined. It involves using augmented matrices and row operations to achieve a simplified form, namely, the row-echelon form.

This example illustrated how, using core concepts of linear algebra, we transformed a system of two equations into a simpler form to directly find the solutions. Understanding these methodologies paves the way for tackling more complex systems and appreciating the interconnections among different mathematical concepts.