Problem 17 - A Complete Resource Book in Mathematics for JEE Main 2017

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 number of solutions is zero.

Step 1: Write the System of Equations

The given equations are: 1. \( x_2 - x_3 = 1 \) 2. \( -x_1 + 2x_3 = 2 \) 3. \( x_1 - 2x_2 = 3 \). We are tasked with finding the number of solutions to this system of equations.

Step 2: Express Variables in Terms of Others

From equation (1) \(x_2 = x_3 + 1\).Substitute \(x_2\) in equation (3): \(x_1 - 2(x_3 + 1) = 3\) simplifies to \(x_1 = 2x_3 + 5\). We now have \(x_1 = 2x_3 + 5\) and \(x_2 = x_3 + 1\).

Step 3: Substitute Back to Check the Third Equation

Using the expressions from Step 2, substitute into equation (2): \(-x_1 + 2x_3 = 2\) becomes \(- (2x_3 + 5) + 2x_3 = 2\). This simplifies to \(-5 = 2\), which is a contradiction.

Step 4: Conclusion on Number of Solutions

Since we encounter a contradiction, the system of equations does not have any solutions. Hence, the given system of equations is inconsistent.

Key Concepts

System of Equations

A system of equations consists of multiple equations that are solved together. Typically, each equation represents a line or plane in space, and solutions correspond to the points where these lines or planes intersect. Systems of equations can be:

• Consistent: having at least one solution. They intersect at one or more points.
• Inconsistent: having no solution due to parallelism or contradiction, resulting in no intersection.
• Dependent: all equations represent the same line or plane, giving infinite solutions.

The objective is usually to determine how these figures overlap each other to find common solutions.

Inconsistent Equations

An inconsistent system of equations is one where no set of values satisfies all equations simultaneously. This typically arises from a contradiction occurring within the equations. In the exercise provided, after substituting variables, the contradiction arises as:\[ -5 = 2 \]which is clearly false.When such contradictions appear, it implies that the lines represented by the equations do not intersect at any point. In practical terms, systems that lead to false statements after simplification, such as conflicting terms, show that no solution exists across all provided equations.

Linear Algebra Concepts

In linear algebra, solving systems of equations involves understanding their structure through matrices and vectors. This can aid in identifying linearly dependent or independent systems and their solutions, if any: - **Matrix representation:** Any system of linear equations can be represented compactly as a matrix equation. This aids in simplifying and solving complex systems with multiple variables. - **Gaussian elimination and RREF:** These are systematic approaches to simplify equations, bringing them into reduced row echelon form to easily conclude about consistency. Utilizing these concepts allows one to analyze the number of solutions effectively, identifying whether the system is consistent, inconsistent, or dependent.