Problem 49 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The system of equations \(2 x-3 y+6 z-5 t=3, y-4 z+\) \(t=1,4 x-5 y+8 z-9 t=k\) has (A) no solution if \(k \neq 7\) (B) no solution if \(k=7\) (C) infinite solutions if \(k \neq 7\) (D) infinite solutions if \(k=7\)

Step-by-Step Solution

Answer

(D) infinite solutions if \(k = 7\).

Step 1: Analyze the Given System of Equations

The system consists of three equations with four variables: \(x, y, z,\) and \(t\). The equations are given as follows:1. \(2x - 3y + 6z - 5t = 3\)2. \(y - 4z + t = 1\)3. \(4x - 5y + 8z - 9t = k\).Since there are more variables than equations, the system may have infinitely many solutions or may be inconsistent.

Step 2: Check for Consistency of System

To determine when the system has a solution, check the potential inconsistency condition. This involves substituting consistent values and checking when the system cannot be satisfied:- Substitute \(k = 7\) to observe the consistency conditions.By solving, it can be identified if there's a specific condition on \(k\) that leads to no solution.

Step 3: Simplify the System by Elimination

Attempt to eliminate variables to simplify the comparison between equations:- Use Eq. 2 \(y = 4z - t + 1\) and substitute into Eq. 1: \(2x - 3(4z - t + 1) + 6z - 5t = 3\), simplifying gives equation involving \(x\) only.- Simplify further to compare with Eq. 3.

Step 4: Check Specific Values

Comparing equations further, substitute various specific values for \(k\):- Substitute \(k = 7\): Both Eqs 1 and 3 should simplify without contradictions.- Any other \(k eq 7\) should introduce a term that cannot be matched, confirming inconsistency.

Step 5: Conclusion Based on Analysis

Upon checking conditions for \(k = 7\), the simultaneous equations, after involving reduced forms, can potentially satisfy each other indicating solutions exist. For \(k eq 7\), the equations become inconsistent.Thus, the solution is infinite only if \(k = 7\).

Key Concepts

Consistency of Systems

When working with systems of equations, a natural question to ask is whether the system is consistent or inconsistent. A consistent system is one that has at least one solution. In the context of linear equations, this typically means that the equations do not contradict each other.
If the equations within a system represent parallel lines or planes that never intersect, the system is deemed inconsistent as it has no solutions possible.

In the given exercise, the consistency is determined by the variable \(k\). By analyzing different values of \(k\), we can see when the system can possibly have a solution, thus ascertaining its consistency.

• If \(k = 7\), the set of equations hold together harmoniously and do not contradict each other, thus confirming consistency.
• For any other value of \(k\) (\(k eq 7\)), the system loses this balance and becomes inconsistent.

Understanding consistency is crucial, as it tells us whether it's worth searching for a solution or recognizing the need for further adjustment or reevaluation of the given equations.

Infinite Solutions

Sometimes a system of equations can have an infinite set of solutions. This typically happens when equations are dependent, meaning one is just a scalar multiple of another or a combination in more complex cases.

With more variables than equations, as seen in the exercise, there is room for infinite solutions if conditions allow. In the analyzed system, making \(k = 7\) allows all three equations to be satisfied.
This means they align perfectly, allowing for a range of variables \(x\), \(y\), \(z\), and \(t\) which satisfy each equation simultaneously.

• Multiple solutions occur when conditions create a balance within the system, forming a consistent relationship between the variables.
• An infinite number of solutions reflect the presence of free variables, meaning some variables can take multiple values leading back to valid equations.

Infinite solutions imply that there is an entire space of solutions rather than a single point, often forming a line or a plane in higher dimensions.

Variables and Equations

Variables and equations are fundamental in system solving. In algebra, variables depict unknowns we aim to solve for, and equations represent relationships between these variables.

The given system involves four variables: \(x\), \(y\), \(z\), and \(t\), across three equations. This setup hints at certain expected outcomes like dependence or independence of variables.

• More variables than equations often indicate that not all variables can be uniquely determined, possibly leading to infinite solutions.
• Every equation provides an additional constraint on variables, helping define their relationships more precisely.

Understanding the role of variables and equations helps in determining the direction in which to move for solving the system, simplifying when possible and recognizing dependencies.

Linear Algebra

Linear algebra is the mathematical framework underpinning systems of linear equations. It provides methods for solving equations, analyzing matrix structures, and understanding vector spaces.
The exercise at hand, though simple in appearance, is deeply rooted in linear algebra concepts.

• Using linear transformations, such as substitutions, we simplify equations to reveal relationships between variables.
• Elimination, a key tool from linear algebra, helps reduce system complexity by removing variables step-by-step.
• Matrix methods or row operations often provide more organized ways to handle systems, making them easier to analyze for solutions or inconsistencies.

In addition, linear algebra's theoretical components give us insights into possible solutions, helping predict whether systems have single, multiple, or no solutions at all, much like the exercise outlines with \(k\) values.