Problem 131 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If the equations \((b+c) x+(c+a) y+(a+b)=0\) \(c x+a y+b=0\) \(a x+b y+c=0\) are consistent then show that either \(a+b+c=0\) or \(a=b=c\).

Step-by-Step Solution

Answer

In summary, the given system of linear equations \((b+c)x + (c+a)y + (a+b) = 0\), \(cx + ay + b = 0\), and \(ax + by + c = 0\) is consistent if either \(a+b+c=0\) or \(a=b=c\). This was demonstrated by showing that in both cases, we can eliminate one variable and the remaining equations are always true or have the same solution set.

Step 1: Write down the given equations

We are given the following system of linear equations: 1. \((b+c)x + (c+a)y + (a+b) = 0\) 2. \(cx + ay + b = 0\) 3. \(ax + by + c = 0\)

Step 2: Show that \(a+b+c=0\) leads to a consistent system

Assume that \(a+b+c=0\). We can use this assumption to eliminate one of the variables from the system. We can add all three equations together: \((a+b+c)x + (a+b+c)y = -3c - 3b - 3a\) Since \(a+b+c=0\), the equation becomes \(0 = 0\), which is always true. This means that if \(a+b+c=0\), the given system of equations is consistent, because we can eliminate one variable, and the equation that remains is always true.

Step 3: Show that \(a=b=c\) leads to a consistent system

Assume that all coefficients are equal, i.e. \(a=b=c\). Then we can rewrite the given system of equations as follows: 1. \(2ax + 2ay = -2a\) 2. \(ax + ay = -a\) 3. \(ax + ay = -a\) We can see that equation 2 and equation 3 are the same. If we divide both sides of equation 1 by 2, we get: 1. \(ax + ay = -a\) Now we can see that all three equations are the same, which means we have a consistent system since all three equations have the same solution set.

Step 4: Conclusion

We have shown that if either \(a+b+c=0\) or \(a=b=c\), then the given system of linear equations is consistent. Hence, the given condition is true for the system to be consistent.

Key Concepts

Linear Equations

When discussing linear equations, we are referring to mathematical expressions that have a certain structure. These equations form straight lines when graphed on a coordinate plane. Typically, a linear equation looks like this: \( ax + by = c \). In this structure:

• \(a\) and \(b\) are coefficients - numbers that multiply the variables.
• \(x\) and \(y\) are variables - the unknowns that we solve for.
• \(c\) is a constant term, a number on its own.

The power of linear equations lies in their simplicity and predictability. By adding or subtracting them, we can understand the relationships between different quantities, which is very useful in both math and real-world applications.

System of Equations

A system of equations is essentially a collection of multiple equations that share the same variables. For instance, if you come across three equations involving \(x\) and \(y\) like they did in the original exercise, you are seeing a system. The goal is to find values for \(x\) and \(y\) that satisfy all equations simultaneously. In practical terms, you might encounter this when working with problems involving:

• Supply and demand calculations.
• Multiple constraints in optimization.
• Equilibrium conditions in physics and economics.

To solve a system, we can use methods like substitution, elimination, or graphical analysis. Each method aims to pinpoint the solution or demonstrate that no solution exists.

Mathematical Consistency

The idea of mathematical consistency is crucial when dealing with systems of equations. Consistency means that there is at least one set of variable values that satisfies all equations in the system simultaneously. In simpler terms, the equations do not contradict each other.In the original exercise, the consistency of the system is checked by evaluating two conditions:

• Condition one: \(a+b+c=0\) simplifies the entire system into a valid identity (\(0 = 0\)). This shows that no matter what values you choose for the variables, the equation holds true, indicating consistency.
• Condition two: \(a=b=c\) results in transformed equations that are identical, which naturally makes the system consistent since all equations essentially state the same relationship.

In conclusion, mathematical consistency ensures that a system is solvable, and understanding this idea helps prevent running into false or impossible results when solving equations.