Problem 28 - A Complete Resource Book in Mathematics for JEE Main

Question

The equations \(x+y+z=6, x+2 y+3 z=10, x+2 y+\) \(m z=n\) give infinite number of values of the triplet \((x,\), \(y, z\) ) if (A) \(m=3, n \in R\) (B) \(m=3, n \neq 10\) (C) \(m=3, n=10\) (D) None of these

Step-by-Step Solution

Answer

Option (C): \( m=3, n=10 \) gives infinite solutions.

Step 1: Identify the Equations

We have three linear equations: \( x + y + z = 6 \), \( x + 2y + 3z = 10 \), and \( x + 2y + mz = n \). We need to determine if these equations yield an infinite number of solutions based on different values of \( m \) and \( n \).

Step 2: Examine the Condition for Infinitely Many Solutions

For the system to have infinitely many solutions, the third equation must be a linear combination of the first two. This occurs when the coefficients of \( z \) adjust to make the third equation identical or dependent on the previous ones.

Step 3: Substitute \( m = 3 \) and Analyze for Dependence

Substituting \( m = 3 \) into the third equation gives us \( x + 2y + 3z = n \). This equation is identical to the second equation \( x + 2y + 3z = 10 \) when \( n = 10 \).

Step 4: Different Scenarios Discerned

- If \( n = 10 \), the third equation is identical to the second, forming a consistent dependent system, hence infinitely many solutions (Case C). - If \( n eq 10 \), the third equation differs and affects the independence of the equations, resulting in a unique solution or no solution (Case B). - For any value of \( n \) when \( m eq 3 \), the equations don't align for infinite solutions, making (Case A) incorrect.

Key Concepts

Systems of Equations

When you work with linear algebra, you often encounter what are called systems of equations. This happens when you have more than one equation that you are trying to solve at the same time. These equations are typically uniting multiple unknown variables. The goal is to find the values of these variables that make all the equations true simultaneously. For instance, if you have a system of two equations like

• \( x + y = 10 \)
• \( x - y = 2 \)

you are looking for values of \(x\) and \(y\) that satisfy both equations.

In our case, the problem involves:

• three equations: \( x + y + z = 6 \), \( x + 2y + 3z = 10 \), and \( x + 2y + mz = n \).

Knowing how to manage such systems is crucial, as they can sometimes have one solution, no solution, or an infinite number of solutions. Understanding the conditions for each scenario helps in determining the nature of the solutions.

Infinite Solutions

In systems of equations, the scenarios sometimes arise where there are infinitely many solutions. This usually occurs when the equations describe the same line or plane. Imagine if you have three equations, and the third one is really the same as a combination of the first two.

For our particular set of equations, infinite solutions happen if the third equation is a perfect overlap with the second one, for example, if we modify its terms so they match. In simpler terms, they need to describe the same path or area in their graphical representations.

In the given exercise, when \(m=3\) and \(n=10\), the third equation becomes identical to the second, \(x + 2y + 3z = 10\). This means every solution of the second equation is also a solution for the third. So, there's not just one point that meets all conditions; rather, many do, forming an infinite set of solutions.

Linear Combination

The idea of a linear combination is fundamental in linear algebra and directly applies to finding infinite solutions in systems of equations. A linear combination is created when you multiply equations by constants and add them together.

In terms of applications, if you can represent one of the equations in a system as a linear combination of others, this signifies dependency among them. In our exercise, for the third equation \(x + 2y + mz = n\), if by setting \(m=3\) and \(n=10\) it becomes exactly the same as the second equation, this depicts a linear combination.

Here’s the trick to spot: Adjust your coefficients (like making \(m=3\)) so that it perfectly matches another equation. When this happens, the solution set expands dramatically, leading to that intriguing case of infinite solutions where each point on a line (or plane) fits the system.