Linear Equations and the Simplex Method

In the context of a system of linear equations, a consistent system is one that has at least one solution, meaning the equations intersect or coincide. An inconsistent system has no solution, meaning the equations are parallel and do not intersect. A dependent system has infinitely many solutions, meaning the equations are equivalent or one equation is a multiple of the other. An independent system has a unique solution, meaning the equations are not equivalent and do not coincide or intersect.

The objective function in the given problem is to maximize the expression $x(x, y) = 7x + y$. The constraints are given by the equation $3x + 4y = -2$ and the non-negativity constraints $x, y \geq 0$.

The simplex method is an algorithm used to find the optimal solution to a linear programming problem. It systematically explores the feasible region by moving from one vertex to another until the optimal solution is reached. The method maximizes or minimizes the objective function while satisfying the given constraints.