Problem 17 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of solutions of equations \(x_{2}-x_{3}=1,-x_{1}+\) \(2 x_{3}=2, x_{1}-2 x_{2}=3\) is (A) zero (B) one (C) two (D) infinite

Step-by-Step Solution

Answer

The system of equations has zero solutions.

Step 1: Write the System of Equations

The given equations are:1. \(x_2 - x_3 = 1\)2. \(-x_1 + 2x_3 = 2\)3. \(x_1 - 2x_2 = 3\)

Step 2: Use Substitution or Elimination Methods

First, solve equation 1 for \(x_2\):\(x_2 = x_3 + 1\).Substitute \(x_2 = x_3 + 1\) in equation 3 to express \(x_1\) in terms of \(x_3\).

Step 3: Solve for \(x_1\)

Substitute \(x_2 = x_3 + 1\) into the third equation:\(x_1 - 2(x_3 + 1) = 3\)Simplify:\(x_1 - 2x_3 - 2 = 3\)\(x_1 - 2x_3 = 5\).

Step 4: Compare Modified System

Now compare the modified equations:Equation 2: \(-x_1 + 2x_3 = 2\)New equation from 3: \(x_1 - 2x_3 = 5\)Add these two equations: \(0 = 7\), a contradiction.

Step 5: Determine the Number of Solutions

Since the resulting equation \(0 = 7\) is a contradiction, it indicates the system of equations is inconsistent. Therefore, there are no solutions.

Key Concepts

Substitution Method

The substitution method is a way to solve a system of linear equations. It involves solving one equation for one variable and then substituting this expression into the other equations. This method simplifies the system by reducing the number of equations and unknowns.

Here’s a simple way to understand substitution:

• First, isolate one variable in one of the equations. In our case, equation 1 (\(x_2 - x_3 = 1\)) is chosen, and we solve for \(x_2\). This gives \(x_2 = x_3 + 1\).
• Next, substitute this expression into the other equations. For example, in equation 3 (\(x_1 - 2x_2 = 3\)), substitute \(x_2 = x_3 + 1\).
• This substitution allows us to express \(x_1\) in terms of \(x_3\). After simplifying, we find that \(x_1 - 2x_3 = 5\).

By reducing the problem to fewer variables, the substitution method simplifies the search for solutions, leading us to easily find any inconsistencies or solutions that exist.

Elimination Method

The elimination method is another technique to solve systems of linear equations. It focuses on removing one variable by adding or subtracting equations, making it possible to solve for the remaining variables.

Here's a step-by-step process:

• Choose two equations from your system. In this case, we have equation 2 (\(-x_1 + 2x_3 = 2\)) and the modified equation 3 (\(x_1 - 2x_3 = 5\)).
• Add these equations together. This step helps eliminate one of the variables. For our example, adding the equations \(-x_1 + 2x_3 = 2\) and \(x_1 - 2x_3 = 5\) results in \(0 = 7\).
• Notice any contradictions or new discoveries about the system’s consistency. The contradiction, \(0 = 7\), signals that the system is inconsistent, meaning no solutions exist for these equations.

The elimination method is powerful for quickly spotting inconsistencies and can greatly simplify finding solutions when applicable.

Inconsistent System

An inconsistent system of linear equations is one where no set of values for the variables can satisfy all the equations simultaneously. In simpler terms, the equations contradict each other, making it impossible to find a common solution.

Understanding the concept:

• Recognize an inconsistency by searching for contradictions. In our previous steps, after applying substitution and elimination, we encountered the contradiction \(0 = 7\).
• This contradiction implies that the equations are not logically compatible. They aim to describe conditions that can't simultaneously be true.
• Such scenarios usually emerge when two lines represented by the equations are parallel and never intersect in a two-dimensional plane.

Hence, whenever you identify that your modified system results in something like \(0 = 7\), you can confidently conclude that the system is inconsistent, meaning it has zero solutions.