State the conditions that a system of three equations in two variables must satisfy in order to

step 1 Here in this problem we have to give the condition such that the system of three linear equation

State the conditions that a system of three equations in two...

Key Concepts

Graphical Representation

Graphically, the condition for a unique solution is illustrated by plotting the three lines represented by the equations. The correct configuration is achieved when all three lines intersect at the same single point. This visual approach helps verify that the system is consistent and that the third equation does not conflict with the intersection point determined by the first two.

Redundancy in Equations

The third equation in an overdetermined system needs to be redundant, meaning it does not provide new information but instead is a linear combination of the other two independent equations. This redundancy confirms that while the system appears to have more equations than unknowns, the essential constraint is maintained by only two independent equations, making the system solvable with a unique solution.

Overdetermined System

A system with more equations than unknowns is termed overdetermined. In such systems, although there are extra equations, a unique solution can still exist if all the equations are consistent with each other. The extra equations must not introduce any contradictions and often turn out to be redundant, meaning they are merely different representations of the same constraint provided by a subset of the equations.

Consistency and Uniqueness

For a system of three equations in two variables to have exactly one solution, all equations must be consistent: they must share exactly one common point of intersection. The first two independent equations determine the unique intersection point, and the third equation must pass through this point as well, ensuring that no inconsistency is introduced.

Intersection of Lines

In a two-variable linear system, each equation represents a line in the plane. Having a unique solution requires that these lines intersect at one and only one point. This means that no two lines can be parallel, unless one is redundant, and all three lines must coincide at the single point of intersection, ensuring agreement among the equations.

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