Problem 20 - College Algebra

Question

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{rr|r}{-2} & {0} & {1} \\ {0} & {2} & {-1}\end{array}\right] $$

Step-by-Step Solution

Answer

\(x = -\frac{1}{2}, y = -\frac{1}{2}\).

Step 1: Understand the Augmented Matrix

The given matrix is an augmented matrix for a system of linear equations. Each row represents an equation, and the last column represents the constants on the right-hand side of the equations. The system of equations corresponding to the given matrix is: \(\begin{align*}-2x + 0y &= 1 \0x + 2y &= -1\end{align*}\).

Step 2: Solve the First Equation for One Variable

The first equation is \(-2x = 1\). To solve for \(x\), divide both sides by \(-2\): \[x = \frac{1}{-2} = -\frac{1}{2}\].

Step 3: Solve the Second Equation for the Second Variable

The second equation is \(2y = -1\). To find \(y\), divide both sides by 2: \[y = \frac{-1}{2}\].

Step 4: Verify the Solution

Substitute \(x = -\frac{1}{2}\) and \(y = -\frac{1}{2}\) back into the original system of equations. For the first equation: \(-2(-\frac{1}{2}) + 0 = 1\), correct. For the second equation: \(0 + 2(-\frac{1}{2}) = -1\), also correct. Both solutions satisfy the original equations.

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool for solving systems of linear equations. It combines both the coefficients of the variables and the constants from the equations into a single matrix format. This structure helps simplify the process of applying methods like Gaussian elimination.

In our exercise, consider the matrix:

• The left portion holds the coefficients:
  • First row: -2 and 0
  • Second row: 0 and 2
• The right portion (after the line) holds the constants:
  • First row: 1
  • Second row: -1

By representing the system visually in this form, you can clearly see and manipulate the essentials of the equations without getting bogged down in excessive notation.

Each row denotes an equation, which means that the augmented matrix portrays our set of linear equations compactly.

Linear Equations

Linear equations are fundamental in mathematics as they describe a straight line and consist of variables raised only to the first power. In the context of our exercise, we have two linear equations:

• The first equation: -2x + 0y = 1 stands for a simple algebraic equality representing a straight line in a 2D plane.
• The second equation: 0x + 2y = -1, similarly represents another straightforward linear relationship.

Solving these equations involves finding the values of the variables (in this case, x and y) that make both equations true simultaneously. This is effectively identifying where the two lines intersect.

The system is represented by the augmented matrix, simplifying the process by reducing the number of steps needed to solve the equations, thus making methods like Gaussian elimination more straightforward.

Solution Verification

After obtaining solutions using Gaussian elimination, it is crucial to verify these solutions to ensure they are correct. This involves substituting the values back into the original equations to see if they satisfy them.

• For the first equation -2x + 0y = 1:
  • Substitute x = -\(\frac{1}{2}\) and y = -\(\frac{1}{2}\). This results in -2(-\(\frac{1}{2}\)) + 0(−\(\frac{1}{2}\)) = 1, which confirms x's accuracy.
• For the second equation: 0x + 2y = -1,
  • Substitute these solutions to obtain 0(-\(\frac{1}{2}\)) + 2(-\(\frac{1}{2}\)) = -1, reaffirming y's correctness.

Verification ensures that each step and calculation are error-free. It is essential as it validates the solution process and confirms that you have accurately solved the system, giving you confidence in the results.