Question

Determine whether the system of equations is in row-echelon form. Justify your answer. $$ \left\\{\begin{aligned} x-9 y+z &=22 \\ 2 y+z &=-3 \\ z &=1 \end{aligned}\right. $$

Short answer

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

Step-by-step solution

Identify the Order of Variables in All Equations

In an equation, the order of appearance should be from left to right as: variable with highest order (like \(x\)) first, then variables with lower order (like \(y\), \(z\)), and at last, the constant (b). In the given system, the order of appearance of variables in each equation is already in the form: \(x\), \(y\), \(z\), b. Which is correct.

Check the Position of Leading Coefficients

The second criteria is to check that each leading coefficient of a non-zero row is in a column to the right of the leading coefficient of the row above it. Here, the first non-zero element of the first equation is \(x\), of the second equation is \(y\), and of the third equation is \(z\). They are following the correct order with respect to the earlier discussion about variables. Hence, this condition is satisfied.

Check for Zeros Below Leading Coefficients

The last criteria is to ensure that all elements below a leading term (that is not a leading term itself) is zero. As there are no such terms in the given system, this condition does not apply.

Conclusion

To conclude, as all the criteria to exist in row-echelon form are satisfied by the given system, it can be proclaimed that the system of equations is indeed in row-echelon form.

Key concepts

Understanding Systems of Equations

A system of equations is an interrelated set of equations, each containing multiple variables. These equations represent relationships between variables, and the goal is often to find the value of each variable that makes all of the equations true simultaneously.

For example, the system of equations:
\[\begin{aligned} x - 9y + z &= 22 \ 2y + z &= -3 \ z &= 1\end{aligned}\]
is composed of three separate equations involving the variables x, y, and z. In a classroom setting, solving such systems teaches students important concepts like substitution, elimination, and the relationship between variables in multiple dimensions.

To solve systems effectively, one helpful method is transforming the system into row-echelon form. The step-by-step analysis showcases how this is checked and why it matters. By reducing equations to a form where they can be simplified and solved more readily, students learn a critical strategy for approaching algebraic problems.

Identifying Leading Coefficients

Leading coefficients are the first non-zero numbers in each equation of a system when written in standard form. They play a pivotal role in solving the system through methods like Gaussian elimination.

For instance, in our system:\[\begin{aligned} x - 9y + z &= 22 \ 2y + z &= -3 \ z &= 1\end{aligned}\]
the leading coefficients are 1, 2, and 1 for x, y, and z respectively. In row-echelon form, each leading coefficient should be to the right of the one in the row above. This staggered arrangement allows us to apply elimination methods to solve for the variables systematically. Focusing on leading coefficients ensures that each step in the process is clear, methodical, and leads us closer to finding the variable values that solve the system.

The Algebraic Order of Variables

The algebraic order of variables is their arrangement based on their powers and degree within an equation, typically from highest to lowest. In a system of equations meant for row-echelon form, it is crucial that variables appear in a consistent and descending order across equations.

In the given system:\[\begin{aligned} x - 9y + z &= 22 \ 2y + z &= -3 \ z &= 1\end{aligned}\]
this order is adhered to with x first, then y, and finally z. This consistency across equations helps maintain a coherent structure for solving. It allows students to clearly see the relationship among variables and facilitates simpler solutions, especially when using strategies like back-substitution to determine the values of variables after the system has been simplified or solved.