Question

Which statement describes the system of equations represented by \(\left[\begin{array}{rrr|r}1 & 5 & -2 & 3 \\ 0 & 1 & 3 & -2 \\ 0 & 0 & 0 & 5\end{array}\right] ?\) (a) The system has one solution. (b) The system has infinitely many solutions. (c) The system has no solution. (d) The number of solutions cannot be determined.

Short answer

The system has no solution (c).

Step-by-step solution

Understand the Matrix Representation

The given matrix is an augmented matrix representing a system of linear equations. The left side of the vertical bar (|) represents the coefficients of the variables, and the right side represents the constants.

Analyze the Rows of the Matrix

Look at each row of the matrix: 1. The first row is \[1, 5, -2 | 3 \]. This represents the equation: \[x_1 + 5x_2 - 2x_3 = 3 \].2. The second row is \[0, 1, 3 | -2 \].This represents the equation: \[x_2 + 3x_3 = -2 \]. 3. The third row is \[0, 0, 0 | 5 \].

Identify the Inconsistency

The third row \[0, 0, 0 | 5 \] indicates that \[0 = 5 \], which is clearly false. This inconsistency means that the system of equations has no solution.

Conclusion

Given the inconsistency in the third row, we conclude that the correct statement about the system is that it has no solution.

Key concepts

augmented matrix

An augmented matrix is a compact and systematic way to represent a system of linear equations. It combines the coefficients of the variables and the constants from the right-hand side of the equations into one matrix.
Think of it as a straightforward method to capture all the critical data of your equations in a single glance.

The augmented matrix provided in the exercise is: \(\begin{bmatrix}1 & 5 & -2 &|& 3 \ 0 & 1 & 3 &|& -2 \ 0 & 0 & 0 &|& 5\thinspace\text{ }\thickspace \thinspace \text{ }\thickspace\thinspace \text{ }\thickspace \thinspace \thinspace\text{ }\firstspace\firspace\firspace\firstspace \firstspace \firstspace\text{ }\thickspace \firstspace\thickspace\firstspace \firstspace\firstspace \firspace\firspace space \firstspace \tistspace space\thickness \thickness \thickness \thinspace \thickspace\thickness \firspace \end{bmatrix}\)
In this matrix:
- The first three columns represent the coefficients of the variables \((x_1, x_2, x_3)\).
- The fourth column, set apart by a vertical bar, represents the constants you want the equations to equal.

It makes analyzing and solving systems of equations easier since you can use techniques like row reduction to reveal key properties of the system.

system of equations

A system of equations is a set of equations with multiple variables that you want to solve simultaneously.
Each equation in the system provides a relationship between the variables.

For example, the first row of the given matrix can be written as: \(x_{1} + 5x_2 - 2x_3 = 3\)
Similarly, the second row is: \(x_2 + 3x_3 = -2\)
To solve a system of equations, you need to find the values of the variables that make all the equations true at the same time.

These solutions can be finite, infinite, or nonexistent, depending on the relationships between the rows of the matrix.

inconsistency in linear algebra

In linear algebra, an inconsistency in a system of equations means that there is no set of variable values that can satisfy all the equations simultaneously.
This usually happens when you get a row in the augmented matrix that translates into a statement that is mathematically impossible.

In our exercise, the last row of the matrix turns into the equation \(0 = 5\), which you can see is a contradiction since zero can never equal five.
This inconsistency indicates that the original system of equations is flawed in such a way that no solutions exist that satisfy all the equations.

Recognizing inconsistencies is crucial, as it lets you know early on that searching for solutions is futile.

no solution in linear systems

When a system of linear equations has no solution, it is referred to as being inconsistent.
This means that there is no possible combination of values for the variables that will satisfy all the equations in the system at once.

This situation arises when the equations represent parallel lines in two dimensions, or parallel planes in three dimensions, that never intersect.
In our exercise, this is precisely the case due to the contradictory row \(0 = 5\).

Recognizing when a system has no solution can save you a lot of effort, as you won't waste time trying to find a solution that doesn't exist.
So always look for inconsistencies in your augmented matrix before proceeding with further calculations.