Problem 142 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what values of \(m\) does the system of equations \(3 x+m y=m\) \(2 x-5 y=20\) has a solution satisfying the conditions \(x>0, y>0\).

Step-by-Step Solution

Answer

The value of \(m\) for which the system of equations has a solution satisfying \(x>0, y>0\) is when \(m>10\).

Step 1: Substitute the equation

Substitute the 1st equation from \(3x+my=m\) to \(x=(m-my)/3\) and we have \(y< m/3\) and \(y>0\).

Step 2: Substitute in the second equation

Substitute \(x\) in terms of \(y\) and \(m\) from step 1 into the 2nd equation \(2x-5y = 20\), this will give us a quadratic equation and solve for \(y\) in terms of \(m\). We will get \(y=(40 + 5m) / (2m+15)>0\).

Step 3: Find the range of m

From Steps 1 and 2, we know that \(y>0, y10\).

Key Concepts

Understanding Inequalities in Equations

Inequalities are mathematical expressions related by symbols that indicate how one value compares to another.When solving problems involving inequalities, you attempt to find a range of values that satisfy specific conditions.In the given problem, we're dealing with inequalities like \(x > 0\) and \(y > 0\).
These conditions ensure that both \(x\) and \(y\) are positive numbers.
Here's what you need to know about the inequalities present in this system:

• The inequality \(y < m/3\) is derived from substituting in one of the equations.
• The inequality \(y > 0\) ensures the variable is a positive real number.

By combining the above inequalities, you find the permissible range for the variables, ensuring they fit all given conditions.

Exploring Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\).They typically have two solutions, which are influenced by the values of \(a\), \(b\), and \(c\).Quadratic equations can often be solved by factoring, completing the square, or using the quadratic formula.
In our problem, substituting one variable from the first equation into the second gives us a quadratic equation in terms of \(y\) and \(m\).

• The structure of this equation is sensitive to the values of \(m\), which affects the solutions for \(y\).
• Solving the equation enables us to express \(y\) in terms of \(m\), helping to fulfill one of the key conditions: \(y > 0\).

Understanding how to manipulate and solve quadratic equations is key to finding valid solutions within specified conditions.

Ensuring Solution Conditions Are Met

The solution conditions state that both \(x\) and \(y\) must be greater than zero.These conditions ensure that our solutions are not just mathematically correct but meaningful for the problem context.
After solving the system and obtaining expressions for \(x\) and \(y\), we check that these satisfy the inequality conditions correctly:

• From the inequality \(y < m/3\), combined with \(y = \frac{40 + 5m}{2m + 15} > 0\), helps determine the valid range of \(m\).
• After simplifying these inequalities, finding \(m > 10\) ensures that both \(x\) and \(y\) remain positive values.

These solution conditions are critical because they provide the feasible region for the values of \(m\), ensuring our system of equations has a valid and understandable solution.