Problem 281 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Solve \(3 x+2 y+7 z=0\) \(4 x-3 y-2 z=0\) \(5 x+9 y+23 z=0\)

Step-by-Step Solution

Answer

The solution is x=0, y=0, z=0

Step 1: Arrangement of Equations

The given system of equations are: \(3 x+2 y+7 z=0\) (1), \(4 x-3 y-2 z=0\) (2), and \(5 x+9 y+23 z=0\) (3)

Step 2: Elimination Method

Multiply Equation (1) by 3 and Equation (2) by 2, this will allow elimination of one variable. The result will be: \(9 x+6 y+21 z=0\) (1'), and \(8 x-6 y-4 z=0\) (2'). Add (1') and (2') together to eliminate y, giving: \(17 x+17 z=0\) or \(x+z=0\) (4)

Step 3: Substitute Equation (4) into (1)

This will permit simplification for a 2 variable equation. Substituting \(x= -z\) will give \(3(-z) + 2y + 7z =0\), which simplifies to \(4z + 2y =0\) or \(2z + y =0\)

Step 4: Solve for y in terms of z

Solving for y in \(2z+ y =0\) we get \(y=-2z\)

Step 5: Substitute Equation for y into (2)

Substitute \(y=-2z\) into the equation \(4 x-3 y-2 z=0\) gives \(4x +6z - 2z =0\), which simplifies to \(4x + 4z =0\) or \(x + z =0\)

Step 6: Compare and find values

Equation 5 is similar to equation 4, and since they must equal each other, this means that x= -z. So substituting \(x = -z\) into \(x + z =0\) we get \(-z + z =0\), solving we find z=0. Substituting z=0 into \(y=-2z\) we find y=0 and similarly for x we find x=0

Key Concepts

Elimination Method

The elimination method is a staple tool in solving systems of linear equations. It involves combining equations to cancel out one or more variables, thereby simplifying the system to one that's easier to manage.

In our exercise, equations are strategically multiplied by constants to create opposites of one variable when the equations are added together, allowing for its elimination. This is a deliberate step towards simplifying the complex system into a set of simpler ones. For instance, by multiplying the first and second equations and then adding them, we eliminated 'y', creating an equation that involved only 'x' and 'z'.

It's crucial for students to learn this method early on, as it not only aids in algebraic solutions for school-level mathematics but also becomes integral for competitive exams like the IIT JEE.

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Key Tips:

• Always look for the best pair of equations to eliminate a variable easily.
• Be mindful of the signs; incorrect signs can lead to wrong elimination.
• Consistency in methodically arranging and solving can prevent errors.

Algebraic Solutions

Algebraic solutions involve finding the set of values that satisfy all equations within a system. In our exercise, we took calculated steps, which included the elimination method, to reduce the system to simpler algebraic forms.

After elimination, we were able, through substitution, to find that the only possible solution was when all variables equaled zero. This is an example of a unique solution, but it's worth noting that systems can also have infinitely many solutions or none at all, emphasizing the interesting nature of algebra.

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Strategy for Success:

• Substitute with care to avoid changing the structure of the equalities.
• Always double-check solutions by plugging them back into the original equations.
• Understand that not all systems will have a straightforward solution like in this case.

IIT JEE Mathematics Preparation

When gearing up for the IIT JEE, a profound grasp of solving systems of linear equations is crucial. The elimination method, as applied in this exercise, is a topic that often appears in the IIT JEE mathematics syllabus.

Aspirants must focus on developing a strong conceptual foundation to tackle a variety of complex problems efficiently. Exercises such as these serve dual purposes: understanding core algebraic concepts and building speed and accuracy required for such competitive exams.

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Preparatory Tips:

• Regular practice with varying difficulty levels of equations is key.
• Time management during practice sessions can help simulate the real exam scenario.
• Understanding the underlying concepts instead of just memorizing steps ensures a deeper understanding and flexibility in solving different problems.