Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the system of equations is in row-echelon form. Justify your answer. $$ \left\\{\begin{array}{rr} x+3 y-7 z= & -11 \\ y-2 z= & -3 \\ z= & 2 \end{array}\right. $$

Step-by-Step Solution

Answer

Yes, the system of equations is in row-echelon form.

Step 1: Check Criteria 1

All rows are non-zero, which means that there are no rows of all zeros. This will satisfy the first criteria.

Step 2: Check Criteria 2

Looking at the leading coefficients (the ones attached to x, y, and z), they are 1 for all rows. This fulfills the second criteria.

Step 3: Check Criteria 3

The leading coefficient in the second row (which is attached to y) is to the right of the leading coefficient in the first row (which is attached to x). Similarly, the leading coefficient in the third row (attached to z) is to the right of the leading coefficient in the second row. This satisfies criteria 3.

Step 4: Check Criteria 4

As visible, columns containing the leading 1 (coefficient) make a consecutive sequence. This, however, is not necessarily a requirement.

Key Concepts

System of Equations

When tackling a system of equations, you are dealing with a collection of two or more equations that have a set of variables. The goal is to find a common solution to these equations, meaning a set of values for the variables that make all equations true simultaneously.

In the context of the provided exercise, we are looking at a system consisting of three linear equations. A linear equation is one that represents a straight line when plotted on a graph. These lines can intersect in various ways, providing us with solutions. If the lines intersect at a single point, they have one unique solution. If they coincide completely, then they have infinitely many solutions; if they are parallel and never meet, the system has no solutions. For the equations given, this system indeed has a potential solution because each equation can be seen as representing a plane in a three-dimensional space, and we are seeking the point or points at which these planes intersect.

Leading Coefficients

The concept of leading coefficients is essential in linear algebra, especially when discussing row-echelon form. A leading coefficient is the first non-zero number in a row of a matrix or an equation. These leading coefficients play a crucial role in determining the nature of solutions to a system of equations.

In our exercise, we observed that the leading coefficients are all '1', which is ideal as it simplifies the process of determining the solutions. It's worth noting that the leading coefficients help us navigate through the technique of row reduction, which is utilized to convert a system of linear equations into a simplified version, such as row-echelon form. The emphasis on leading coefficients being '1' ensures that each variable can be solved incrementally, a step that is critical in understanding the relationship between the equations in the system.

Linear Algebra

Diving deeper, linear algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. It's a foundation for advanced areas of mathematics and applications in science and engineering.

In terms of solving systems of linear equations, linear algebra provides tools such as the row-echelon form, which is a pivotal concept demonstrated in our exercise. The row-echelon form simplifies the system to a stage where the solution can be found through back substitution. Our system, as verified in the steps, is already in this desirable form since it adheres to specific patterns such as having leading coefficients aligned in a top-left to bottom-right fashion. The ability to recognize and utilize row-echelon form allows for efficient and systematic methods of finding solutions to even the most complex systems of equations.