Question

Show that the system $$ \left\\{\begin{array}{r} x+2 y-z=a \\ 2 x+y+3 z=b \\ x-4 y+9 z=e \end{array}\right. $$ is inconsistent unless \(c=2 b-3 a\).

Short answer

The system is inconsistent unless \(c = 2b - 3a\).

Step-by-step solution

Write the System in Augmented Matrix Form

To solve the system, first represent it as an augmented matrix. The augmented matrix is obtained by aligning the coefficients of each variable in rows, followed by the constant from each equation. The corresponding augmented matrix is:\[\begin{bmatrix}1 & 2 & -1 & | & a \2 & 1 & 3 & | & b \1 & -4 & 9 & | & e\end{bmatrix}\]

Perform Gaussian Elimination

Apply row operations to make the matrix into row-echelon form. First, substract row 1 from row 3 to eliminate the first element of row 3:\[\begin{bmatrix}1 & 2 & -1 & | & a \2 & 1 & 3 & | & b \0 & -6 & 10 & | & e-a\end{bmatrix}\]Next, substract two times row 1 from row 2 to eliminate the first element of row 2:\[\begin{bmatrix}1 & 2 & -1 & | & a \0 & -3 & 5 & | & b-2a \0 & -6 & 10 & | & e-a\end{bmatrix}\]

Further Simplify the Matrix

Notice that row 3 can be made scalar multiple of row 2, allowing further simplification. Divide row 3 by 2:\[\begin{bmatrix}1 & 2 & -1 & | & a \0 & -3 & 5 & | & b-2a \0 & -3 & 5 & | & \frac{e-a}{2}\end{bmatrix}\]Now subtract row 2 from row 3 to obtain:\[\begin{bmatrix}1 & 2 & -1 & | & a \0 & -3 & 5 & | & b-2a \0 & 0 & 0 & | & \frac{e-a}{2}-(b-2a)\end{bmatrix}\]

Determine the Condition for Consistency

For the system to be consistent, the right-most column in the third row, which represents the zero row, must equal zero:\[0 = \frac{e-a}{2}-(b-2a)\]Simplify the expression:\[0 = \frac{e-a}{2} - b + 2a\]\[0 = \frac{e-a - 2b + 4a}{2}\]\[0 = e + 3a - 2b\]Thus, rearrange it to show the condition for consistency is:\[e = 2b - 3a\]

Conclusion on System's Consistency

The system is inconsistent unless the variable equality condition derived above is satisfied. Therefore, the system can only be consistent if:\[c = 2b - 3a\]

Key concepts

Row-Echelon Form

The concept of **Row-Echelon Form** is fundamental in linear algebra when solving a system of linear equations using Gaussian Elimination. The Row-Echelon form of a matrix is a staircase-looking form where each leading entry (also known as a pivot) in a row is 1, and it is to the right of any pivot in any row above it. Additionally, all entries below a pivot are zeros. This structure simplifies the matrix and makes it easier to solve systems of equations.

Consider a matrix where the rows have been adjusted using row operations: - **Swapping** rows ensures the pivot is in the correct position. - **Multiplying** rows by non-zero constants can simplify the numbers. - **Adding/Subtracting** multiples of rows from one another helps make entries below the pivot zero.

The main advantage of transforming a matrix into its Row-Echelon Form is that it makes back-substitution possible, allowing us to find the solutions to the system efficiently. In our particular example, reaching the Row-Echelon Form was necessary to deduce the conditions for system consistency.

Augmented Matrix

An **Augmented Matrix** is a powerful tool in linear algebra to represent a system of linear equations compactly and efficiently. It combines both the coefficient matrix and the constants from the equations into one single matrix. This not only allows for simpler manipulation but also facilitates the process of performing Gaussian Elimination.

To form an augmented matrix from a system of equations, align the coefficients of the variables and the constant terms clearly, using a vertical bar to separate the coefficient part from the constant part. In our exercise, the augmented matrix is represented as:\[\begin{bmatrix}1 & 2 & -1 & | & a \2 & 1 & 3 & | & b \1 & -4 & 9 & | & e\end{bmatrix}\]

This matrix condenses the given system of equations into a format that is easy to work with, making it simpler to apply row operations to solve for the variables involved. By transforming the augmented matrix, we gain insights into the properties of the system, like whether a solution exists, and under what conditions it might be consistent or inconsistent.

System of Linear Equations

A **System of Linear Equations** comprises multiple linear equations, attempting to find common solutions for the variables involved. Representing these equations using matrices simplifies the solution process through structured operations like Gaussian Elimination.

In a typical system, equations are paired together, each contributing relationships among the variables. Solving such systems might result in: - A **unique solution**, if the system has exactly one intersection point. - **Infinitely many solutions**, when equations represent the same plane or line. - **No solution**, indicating that the equations define parallel planes or lines that never intersect.

In our exercise, we analyzed a given system of three equations with three unknown variables. Through the manipulation into row-echelon form, we identified the conditions making the system consistent or inconsistent. Determining the specific values of constants—whether the system's equations align or contradict each other—helps in comprehending the scope and nature of its solutions.