91Ó°ÊÓ

Dependent equations in a system are equations that are essentially the same even though they may appear different at first. This is because they derive from each other by multiplication or division by a constant. In the given exercise, notice how equation 3 is a multiple of equation 1. If we multiply equation 1 by 2, it becomes \( 8x - 2y + 6z = 20 \). Ideally, equation 3 should match this result if it were exactly dependent. However, equation 3 is given as \( 8x - 2y + 6z = 10 \), which is incorrect due to the difference in the constant term (20 vs 10).
This discrepancy indicates that there is an error in our assumption about dependency, leading us to realize that the system cannot be dependent in this case as they do not represent the same plane in three-dimensional space. Comprehending dependent equations helps in simplifying systems of equations because if one equation depends on another, they don’t contribute new information for solving the system.

An inconsistent system of equations is one where no single set of variables satisfies all of the equations simultaneously. In our exercise, you've identified the reason as being the mismatch in the equations mentioned previously. The third equation can't be satisfied if it's checked against the possible solutions of the first two.
Being inconsistent often means the equations represent lines or planes that do not intersect at a single point. With two independent equations, combining them should ideally give solutions that satisfy them if consistent. However, if even after valid substitutions and simplifications no coherent result exists, the system is inconsistent. Remembering to check for inconsistent systems can save time spent chasing solutions that do not exist, ensuring that your approach in solving the system is optimal.

The substitution method involves solving one of the system's equations for one variable and then substituting that solution into the other equations. This process simplifies the system, making it easier to solve for the remaining variables. In our system, equation 2 was chosen to solve for \( x \), giving \( x = 5 - y + z \).
This expression for \( x \) is then substituted into equation 1, simplifying the system. The key advantage of substitution is it reduces the number of equations and variables you have to handle at each step. Though manual, it provides clearer tracking of substitutions, reducing errors that can arise in complex manual manipulations.
It’s crucial to pick the equation most easily solvable for a variable to avoid cumbersome algebraic manipulations. If you find the system inconsistent, as in our exercise, substitution exposes it, revealing any contradictions that point towards an inconsistent set of equations.