The equations x+4 y+8 z=16,3 x+2 y+4 z=12 and 4 x+y+2 z=10 have (a) only one solution (b) two so

step 1 So in the given question we have a set of equations that are given us x plus 4y plus 8 z is equa

The equations x+4 y+8 z=16,3 x+2 y+4 z=12 and 4 x+y+2 z=10...

Key Concepts

Matrix Rank and Consistency

The rank of the coefficient matrix (and the augmented matrix) is a fundamental concept for understanding the solution set of a system. The rank helps determine if a system is consistent (has at least one solution) or inconsistent (has no solution) and whether the system is underdetermined, leading to infinitely many solutions.

Linear Dependence

Linear dependence refers to a situation where some equations in the system can be expressed as linear combinations of others. This redundancy affects the uniqueness of solutions, with dependent equations typically leading to either an infinite number of solutions (if they do not conflict) or, in some cases, to inconsistency if combined with contradictory equations.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The main focus when studying these systems is to determine the values of the variables that satisfy all equations simultaneously. This analysis spans topics such as methods of solving, solution types, and the structure underlying the equations.

Unique, Infinite, or No Solutions

Systems of linear equations can yield exactly one solution, infinitely many solutions, or no solution at all. The classification depends on whether the equations intersect at a single point, align to form a plane or line with many intersection points (indicating dependency), or conflict with one another (indicating inconsistency).

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