Problem 20 - College Algebra with Corequisite Support

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

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

Step 1: Write Augmented Matrix

Start with the given system of equations and write it in augmented matrix form. The matrix given is already in augmented form, which is: \[ \left[\begin{array}{rr|r} -2 & 0 & 1 \ 0 & 2 & -1 \end{array}\right] \] This represents the system of equations: \[-2x + 0y = 1\] \[0x + 2y = -1\]

Step 2: Simplify Each Row

Start by simplifying each row if necessary. Row 1 is \([-2, 0, |, 1]\), which can be simplified by dividing by \(-2\): \[R1 = \frac{1}{-2}R1\] This results in: \[R1: [1, 0, |, -\frac{1}{2}]\]. Row 2 is already simple: \[R2: [0, 2, |, -1].\]

Step 3: Interpret the Solution

The transformed matrix is: \[ \left[\begin{array}{rr|r} 1 & 0 & -\frac{1}{2} \ 0 & 2 & -1 \end{array}\right] \] This corresponds to the equations: \[x = -\frac{1}{2}\] \[2y = -1\] Divide the second equation by 2 to solve for \(y\): \[y = -\frac{1}{2}.\]

Step 4: Solution Verification

The final values obtained are \(x = -\frac{1}{2}\) and \(y = -\frac{1}{2}\). Substitute these back into the original equations to verify consistency: For the first equation: \(-2(-\frac{1}{2}) + 0(-\frac{1}{2}) = 1\) This simplifies to: \(1 = 1\) For the second equation: \(0(-\frac{1}{2}) + 2(-\frac{1}{2}) = -1\) This simplifies to: \(-1 = -1\) Both conditions satisfy the original equations, confirming the solution.

Key Concepts

Systems of Equations

In mathematics, a **system of equations** is a set of two or more equations that share two or more unknowns. The goal is to find values for these unknowns that satisfy all equations in the system simultaneously. Typically, systems of equations can appear in different forms, such as linear or nonlinear.

Linear systems are the most common and are composed of linear equations. An example is:

• \(-2x + 0y = 1\)
• \(0x + 2y = -1\)

These can be written as an augmented matrix, which is a compact way of handling system equations using matrices.Each equation in the system should be considered together to find a common solution. The concept of solving such systems often involves methods like **substitution** and **elimination** or using matrix operations, such as Gaussian elimination.

Augmented Matrix

An **augmented matrix** is a powerful tool used to solve systems of equations. It combines the coefficients and constants of a system into a matrix format.
The matrix used in this exercise is:\[\begin{array}{rr|r} -2 & 0 & 1 \ 0 & 2 & -1 \\end{array}\]Each row represents an equation, and each column corresponds to a variable's coefficient or constant term.

The vertical line in the matrix separates the coefficients of the variable from the constants on the right side of the equation.

• The left side: coefficients of variables.
• The right side: constant terms.

Using an augmented matrix simplifies applying operations like row transformations. These operations help reduce the matrix into a simpler form where solutions can be easily interpreted. This method, known as Gaussian elimination, allows us to efficiently solve systems of linear equations by first transforming the system into simpler equivalent equations.

Solution Verification

**Solution verification** is a crucial step in solving systems of equations. After finding values for the unknowns, it’s important to ensure they satisfy all original equations.
This means substituing the found values back into the original set of equations.

In our solution, we obtained:

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

To verify these, substitute into the original system:- **First equation**: \(-2(-\frac{1}{2}) + 0(-\frac{1}{2}) = 1\), which simplifies to \(1 = 1\).- **Second equation**: \(0(-\frac{1}{2}) + 2(-\frac{1}{2}) = -1\), which simplifies to \(-1 = -1\).Both substitutions hold true, showing that the provided solution satisfies the complete system, confirming its accuracy and that the solution is indeed correct.