Problem 67 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the equations \((a+1)^{3} x+(a+2)^{3} y=(a+3)^{3},(a+1) x+(a+2) y\) \(=a+3, x+y=1\) are consistent then \(a\) is equal to (A) 1 (B) \(-1\) (C) 2 (D) \(-2\)

Step-by-Step Solution

Answer

The value of \(a\) is \(-2\).

Step 1: Write down the equations

We begin by writing down the given system of equations: \((a+1)^{3} x + (a+2)^{3} y = (a+3)^{3}\), \((a+1) x + (a+2) y = a+3\), \(x+y = 1\).

Step 2: Express in terms of variables

From equation \(x+y=1\), replace \(y\) with \(1-x\) in the other two equations. This simplifies the system to two variables: \((a+1)^{3} x + (a+2)^{3} (1-x) = (a+3)^{3}\), and \((a+1) x + (a+2)(1-x) = a+3\).

Step 3: Simplify the equations

Substitute \(y = 1-x\) into the equations:1. \((a+1)^{3} x + (a+2)^{3} - (a+2)^{3} x = (a+3)^{3}\)2. \((a+1) x + (a+2) - (a+2) x = a+3\).

Step 4: Solve the second equation

Simplify the second equation: \((a+1)x + (a+2) - (a+2)x = a+3\) which simplifies to \(x = 1\).

Step 5: Substitute x into the first equation

Substituting \(x = 1\) into the first simplified equation results in:\((a+1)^{3} - (a+2)^{3} + (a+2)^{3} = (a+3)^{3}\).

Step 6: Check consistency by solving the algebraic equation

After simplifying, check whether this equality holds:\((a+1)^{3} = (a+3)^{3}\).This implies solving \(a+1 = a+3\), which is impossible unless \(a = -2\) allows consistent solution without contradictions.

Key Concepts

Consistency of Equations

In a system of equations, consistency is crucial. It indicates that there is at least one solution that satisfies all equations in the system. In this case, to determine the value of \( a \) that ensures consistency, we need each equation to align perfectly without any contradictions. When equations are consistent:

• They intersect at common points.
• There is at least one solution.
• All expressions are logically coherent within the context of the equations.

If, for instance, solving leads us to a situation like \( a+1 = a+3 \), we encounter a contradiction unless specific values like \( a = -2 \) make them consistent. Understanding consistency helps us avoid false solutions.

Substitution Method

The substitution method is a useful technique to solve a system of equations. It involves expressing one variable in terms of another and substituting this expression into the other equations. This step leads to equations with fewer variables, simplifying the process of finding solutions. Here's how it works in this scenario:

• From the simple equation \( x+y=1 \), we can express \( y \) as \( 1-x \).
• Substitute \( y = 1-x \) into the remaining equations.
• This results in equations predominantly in terms of \( x \), simplifying the solving process.

By reducing the number of variables, the substitution method makes complex systems more manageable, allowing us to pinpoint solutions more effectively.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to find solutions. It is vital in solving the system of equations and checking their consistency. The general steps include:

• Simplifying expressions by combining like terms and performing operations that maintain equality.
• Substituting values, like \( x = 1 \), into equations to test their validity.
• Breaking complex expressions into simpler components for easier handling.

In our problem, simplifying the equations and substituting \( x = 1 \) was key to determining if the system is consistent. Algebraic manipulation ensures clarity, allowing us to identify conditions, such as \( a = -2 \), for solutions to exist without contradictions.