Problem 73 - Precalculus

Question

Solve the system of linear equations using Gauss-Jordan elimination. $$\begin{array}{r} x+y=4 \\ -3 x-3 y=10 \end{array}$$

Step-by-Step Solution

Answer

The system has no solution; it's inconsistent.

Step 1: Write the System as an Augmented Matrix

To begin, let's convert the system of equations into an augmented matrix form. The system is:\[\begin{array}{r} x + y = 4 \-3x - 3y = 10 \end{array}\]This translates to the augmented matrix:\[\begin{bmatrix}1 & 1 & | & 4 \-3 & -3 & | & 10\end{bmatrix}\]

Step 2: Simplify the Matrix

To simplify, we'll first aim to obtain a leading 1 in the first row. It's already a 1 in position (1,1), so no changes are needed for the first row. Now, transform the second row by adding three times the first row to it to make the element in position (2,1) equal to zero. This gives:\[\begin{bmatrix}1 & 1 & | & 4 \0 & 0 & | & 22\end{bmatrix}\]

Step 3: Interpret the Result

The resulting matrix gives us the transformed system of equations:\[\begin{array}{r}x + y = 4 \0 = 22\end{array}\]The equation \(0 = 22\) indicates a contradiction, meaning the system has no solution.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. In simpler terms, it's a collection of lines on a graph that we aim to find common intersections for. For instance, the system:

• \(x + y = 4\)
• \(-3x - 3y = 10\)

represents two lines in a coordinate plane. Solving the system means finding the point(s) where these lines intersect, which gives the solution(s) for the variables involved. It's important to accurately transform these equations into a matrix format to perform operations more effectively.

Augmented Matrix

An augmented matrix is a compact and efficient way of representing a system of linear equations. It is formed by writing the coefficients and constants of the linear equations into a matrix format. For example, from the system:

• \(x + y = 4\)
• \(-3x - 3y = 10\)

The corresponding augmented matrix is:\[\begin{bmatrix} 1 & 1 & | & 4 \-3 & -3 & | & 10 \end{bmatrix}\]Here, the vertical line separates the coefficients of the variables from the constants on the right side of the equations. This matrix form allows us to apply efficient methods like Gauss-Jordan elimination to find solutions.

Inconsistent System

An inconsistent system is one in which there are no solutions. It generally occurs when the equations represent parallel lines that never intersect. This means the equations are contradictory, like in our example where transforming the matrix leads to:\[\begin{bmatrix} 1 & 1 & | & 4 \0 & 0 & | & 22 \end{bmatrix}\]The zero row with a non-zero constant shows the system has reached a contradiction—0 cannot equal 22. Thus, the system is inconsistent as it suggests the lines would meet in space that simply does not exist in this context.

No Solution

The term no solution refers to the scenario in which a system of linear equations does not have any point of intersection. For the problem we have discussed, the equations describe parallel lines in the graph. Since parallel lines never meet, the system doesn’t have any point (or solution) that satisfies both equations simultaneously. The contradiction evident from the row \([0 \, 0 \, | \, 22]\) implies that no value of the variables \(x\) and \(y\) will ever satisfy both equations at the same time. Thus, it concludes that the system yields no solution, and this is a concrete example of a real-world scenario where different rules or conditions setup cannot be resolved by a single common decision or point.