Problem 63 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

(B) \(m \in (-\infty, \frac{-15}{2}) \cup (30, \infty)\)

Step 1: Understanding the System of Equations

We have two equations:1. \(3x + my = m\)2. \(2x - 5y = 20\)We need to find the value of \(m\) for which this system has a solution satisfying \(x > 0\) and \(y > 0\).

Step 2: Solve the Second Equation for \(x\)

From the second equation, \(2x - 5y = 20\), solve for \(x\):\[x = \frac{5y + 20}{2}\]

Step 3: Substitute into the First Equation

Substitute \(x = \frac{5y + 20}{2}\) from Step 2 into the first equation \(3x + my = m\):\[3\left(\frac{5y + 20}{2}\right) + my = m\]

Step 4: Simplify the Equation

Simplify the above equation:\[\frac{15y + 60}{2} + my = m\]Multiply everything by 2 to eliminate the fraction:\[15y + 60 + 2my = 2m\]

Step 5: Rearrange Terms

Rearrange the equation from Step 4:\[(15 + 2m)y = 2m - 60\]Solve for \(y\):\[y = \frac{2m - 60}{15 + 2m}\]

Step 6: Determine Conditions for \(y > 0\)

For \(y > 0\), the numerator and denominator in \(y = \frac{2m - 60}{15 + 2m}\) must have the same sign:1. \(2m - 60 > 0\) implies \(m > 30\)2. \(15 + 2m > 0\) simplifies to \(m > -\frac{15}{2}\)

Step 7: Determine Conditions for \(x > 0\)

From Step 2, for \(x = \frac{5y + 20}{2} > 0\):This implies \(5y + 20 > 0\) which holds as long as \(y > 0\).

Step 8: Combine Conditions

From Step 6, \(m > 30\) and \(m > -\frac{15}{2}\), combine these conditions with the requirement \(x > 0\). The value of \(m\) that satisfies both is \(m > 30\).

Step 9: Conclusion and Answer

Considering both inequalities satisfies the conditions, the possible interval from the options that includes \(m > 30\) is in option (B):- \(m \in (-\infty, \frac{-15}{2}) \cup (30, \infty)\)

Key Concepts

System of Equations

In mathematics, a system of equations is a collection of two or more equations with the same set of variables. To find a solution for a system means you need to find the variable values that satisfy all the equations in the system simultaneously.

• For instance, in this exercise, the two equations are: \(3x + my = m\) and \(2x - 5y = 20\).
• The solution to the system will be specific values for \(x\) and \(y\) such that both equations are true at the same time.

Key to solving these systems is substitution or elimination methods. Here, substitution was used, where one equation was solved for one variable and substituted back into the other.
Understanding how to manipulate equations and isolate variables is critical for solving such systems. This approach helps you find relationships between variables, making it easier to solve them under given conditions.

Solution Conditions

For a system of equations, solution conditions are the specific requirements or constraints that potential solutions must meet. In the context of this exercise, solutions are valid only if \(x > 0\) and \(y > 0\).

• This means solving the system isn't enough; we must ensure the solutions also satisfy these positivity conditions for both variables.
• To determine the conditions, we substitute expressions to analyze when positive conditions hold true.

First, by rearranging and analyzing the solution for \(y\), we find that \(y = \frac{2m - 60}{15 + 2m}\) must be positive, implying constraints on \(m\). This involves setting the numerator and denominator to the same sign, which results in the inequalities:

• \(2m - 60 > 0\) leading to \(m > 30\).
• \(15 + 2m > 0\) simplifying to \(m > -\frac{15}{2}\).

By combining these constraints, one resolves that \(m > 30\) effectively ensures positive solutions for both \(x\) and \(y\).

Inequality Solving

Inequality solving is a process to find the range of values that satisfy a given inequality. It's similar to solving an equation but with additional considerations due to the inequality signs.
In this exercise, inequalities are essential to ensure the positivity of the solutions. You will encounter expressions such as \(2m - 60 > 0\) and \(15 + 2m > 0\), which will direct you on the possible values of \(m\) that meet the positivity criteria.
Steps to solve inequalities:

• Simplify each inequality to find the critical values. For example, for \(2m - 60 > 0\), add 60 to get \(2m > 60\), then divide by 2 resulting in \(m > 30\).
• Interpret the inequality based on conditions provided: If both \(x > 0\) and \(y > 0\) must hold, combine the restrictions from all inequalities to find a feasible range for \(m\).

Solving inequalities often involves multiple conditions, and understanding these can help dissect each part individually, ensuring comprehensive solutions that fit all criteria set within a problem.