91Ó°ÊÓ

In the context of linear systems, a solution set consists of all possible solutions that satisfy all equations in the system simultaneously. When solving a system of equations, the objective is to find the values of the variables that make every equation true. For example, in system (a), the solution set is expressed as

• \( x_1 = 2x_3 + 2 \)
• \( x_2 = -4x_3 - 1 \)
• \( x_3 = x_3 \)

This means that for any choice of \( x_3 \), you can find the corresponding values of \( x_1 \) and \( x_2 \) that satisfy both equations. This method reflects an infinite solution set parameterized in terms of \( x_3 \). The values are dependent on this parameter, showcasing the flexibility and variety of possible solutions in such systems.

In linear systems, inconsistencies arise when the equations conflict in a way that no solution satisfies all of them simultaneously. This happens when the lines or planes represented by the equations do not intersect. In system (b), after substituting the known variables into the third equation, you end up with a mathematical inconsistency—an equation that contradicts itself:

• Equation becomes: \[ 9 - 3x_3 = -1 \]
• Leading to: \[ 10 = -3x_3 \]
• Result: \[ x_3 = \frac{10}{3} \]

This result suggests that for values other than \( x_3 = \frac{10}{3} \), a valid solution does not exist, indicating a potential error or misalignment in expected outcomes when calculating other variables. Therefore, the system can yield a solution only under specific constraints, which highlights its limited range of applicability.

Equations within a linear system often need transformation to make them more manageable. This process involves isolation of variables, combination, or substitution, as needed. Typically, you start by expressing one variable in terms of another and gradually replace variables until each is isolated and a complete solution can be determined. For example, in system (c), solving begins with expressing \( x_1 \) in terms of the other variables:

• \( x_1 = -1 - 2x_2 + x_3 \)

As you substitute and simplify using this expression in related equations, you narrow down the results to identify consistent answers across equations. Each step grows progressively simpler until you identify all values matching both equations' needs. This methodical approach uncovers the rooted complexity of interacting variables, steering you toward the system's solutions or revealing the absence thereof.

The substitution method can be a powerful technique in solving linear systems, especially when quickly isolating and solving for variables. The process involves expressing one variable from one equation and substituting it into another, allowing you to simplify the system one equation at a time. In system (d), starting by simplifying and solving for \( x_1 \) from a manipulated equation illustrates this:

• Simplify equation: \( 4x_1 + x_2 + 2x_3 = -1 \)
• Isolate \( x_1: \)
• \( x_1 = -\frac{1}{4} - \frac{x_2}{4} - \frac{x_3}{2} \)

Once substituted, the resulting new equation is analyzed, leading to the resolution of the system or discovering inconsistent terms. The technique aids in swiftly narrowing down variables, offering a step-by-step resolution pathway that eventually simplifies to a solvable or cleared system. Using substitution is akin to patiently untying a knot, making each step and calculation pivotal to arriving at a correct solution set.