Understanding Systems of Simultaneous Equations

A system of simultaneous equations involves two or more equations with the same set of variables. The goal is to find values for the variables that satisfy all equations in the system at the same time.

For a system of simultaneous equations involving linear equations, there are typically three possible outcomes regarding the number of solutions:

• Exactly one solution (unique solution).
• No solution.
• Infinite solutions.

Analyzing the Case of Infinite Solutions

When a system of simultaneous equations has infinite solutions, it means that the equations are not truly distinct; they are essentially equivalent or proportional to each other. In the case of two linear equations in two variables, this happens when the graphs of the equations are the exact same line.

Consider a system of two linear equations:

\begin{equation*} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{equation*}

This system has infinite solutions if the coefficients and constants are proportional, meaning $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (assuming denominators are non-zero).

Classification Based on Solutions: Dependent System

The equations classification for a system of simultaneous equations depends on the number of solutions it has. Specifically:

• If there is exactly one solution, the system is called Consistent and Independent.
• If there is no solution, the system is called Inconsistent.
• If there are infinite solutions, the system is called Consistent and Dependent.

The term 'Dependent' in the equations classification signifies that the equations within the system are linearly dependent. One equation can be obtained by multiplying another equation by a non-zero constant or by a linear combination of other equations in larger systems.

Therefore, a system of simultaneous equations with infinite solutions is classified as Dependent.

Based on this, a system having infinite solutions is correctly termed 'Dependent'.