The values of m for which the system of equations 3 x+m y=m...

The values of $m$ for which the system of equations $3 x+m y=m$ and $2 x-5 y=20$ has a solution satisfying the condition $x>0, y>0$, are (A) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(0, \infty)$ (B) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(30, \infty)$ (C) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(0,30)$ (D) None of these

The values of $m$ for which the system of equations $3 x+m y=m$ and $2 x-5 y=20$ has a solution satisfying the condition $x>0, y>0$, are
(A) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(0, \infty)$
(B) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(30, \infty)$
(C) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(0,30)$
(D) None of these

Key Concepts

Positivity Constraints

Positivity constraints specify that the variables in the solution must be greater than zero. This adds a layer of optimization or inequality analysis to the problem, restricting the set of acceptable solutions by imposing additional conditions on the variables.

Inequality Analysis

Inequality analysis is used to derive conditions on the parameters or variables to meet certain requirements, such as the positivity of solutions. This involves determining the ranges of the parameter values that satisfy the inequality constraints imposed by the problem.

System of Linear Equations

A system of linear equations involves multiple equations that share the same set of variables. The goal is to find values for these variables that satisfy all the equations concurrently, often using techniques such as substitution, elimination, or matrix methods.

Parameter-Dependent Systems

In a parameter-dependent system, one or more of the coefficients in the equations include an extra variable (parameter) that influences the nature of the solution. Analysis involves determining for which parameter values the system has a unique solution, infinitely many solutions, or no solution.

Algebraic Manipulation

Algebraic manipulation, including substitution and elimination methods, is essential for simplifying the system of equations. By isolating variables and expressing one in terms of another, the problem is reduced to simpler forms that can be used to analyze constraints and solve for the parameters.

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