Problem 279 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Solve \(3 x-y+z=0\) \(-15 x+6 y-5 z=0\) \(5 x-2 y+2 z=0\)

Step-by-Step Solution

Answer

From the above steps, the solution to the system of equations 1. \(3x-y+z=0\),2. \(-15x+6y-5z=0\) and3. \(5x-2y+2z=0\) is \(x=0\) and \(y = z\). That means any real number could be a solution for y and z.

Step 1: Simplification

First off, simplify the given equations by dividing each equation by their greatest common divisor (gcd), which gives the simplified equations:1. \(3x-y+z=0\), (gcd=1), simplified equation remains the same.2. \(-15x+6y-5z=0\) (gcd=3), simplified equation: \(-5x+2y-2z=0\)3. \(5x-2y+2z=0\) (gcd=1), simplified equation remains the same.

Step 2: Elimination of Variable y

From the simplified equations, notice that equation-2 and equation-3 have the same coefficients but with different signs for y and z, hence we can add equation-2 and equation-3 to eliminate y variable:1. (\(-5x+2y-2z=0\)) + (\(5x-2y+2z=0\)) = \(0x+0y+0z=0\), which simplifies to \(0=0\). This equation is true, but it doesn't provide any information about the variables, so we discard it.

Step 3: Solving for remaining Variables

Since we have only two equations left, and they both consist of the same variables, we can once again use addition to eliminate one of the variables. This time let's eliminate z. By subtracting equation-1 from equation-3, we get: \((5x-2y+2z) - (3x-y+z)\) which simplifies to \(2x-y=0\), or \(y = 2x\). Now, substitute \(2x\) in place of \(y\) in equation 1, we get a single variable equation which we can solve., we get \(3x - 2x + x = 0\), or \(2x=0\). Solving this equation gives \(x = 0\). Substituting \(x=0\) back into equation 1 gives \(3*0 - y + z = 0\), which simplifies to \(y = z\).

Key Concepts

Gaussian Elimination

Gaussian Elimination is a method used to solve systems of linear equations. It involves transforming the system's augmented matrix to its row echelon form or reduced row echelon form through a series of elementary row operations. The main goal is to simplify the equations to make them easier to solve.

To perform Gaussian Elimination, follow these steps:

• Write the system of equations as an augmented matrix.
• Use row operations to get zeros below the leading coefficients (called pivot positions) in each column.
• Continue these operations down the matrix until achieving row echelon form.
• Optionally, further simplify to get reduced row echelon form where needed.

The process transforms the system into a triangular form where the equations can be solved sequentially. This exercise simplifies the matrix through elimination steps, making the equations easier to solve.

Substitution Method

The Substitution Method involves solving a system of equations by expressing one variable in terms of the other(s), and substituting this expression into another equation. This method can simplify a problem to a single equation in one variable, making it easier to solve.

In this exercise, the substitution method was utilized by first reducing the three equations to two by eliminating one of the variables. This was achieved by subtraction of one equation from another. Then, the solution found for one variable was substituted back into a simplified version of another original equation.

For example:

• The equation for y was found in terms of x ( y = 2x
)

• Then, this expression was substituted back into one of the remaining equations to find x

This method is effective, especially when one variable is isolated, allowing gradually reducing equations to fewer variables step by step.

Simplification of Equations

Simplification is the process of rewriting equations in a simpler form, ideally involving smaller integers and simpler expressions. Often, this involves reducing coefficients or eliminating fractions by dividing all terms by the greatest common divisor (GCD).

In this problem, simplification was the first step. Each equation was simplified, if possible, by dividing all terms by their respective GCD. This made the subsequent steps significantly easier:

• The first equation remains unchanged since it was already in its simplest form.
• The second equation was simplified by dividing all terms by 3.
• The third equation was already simplest and required no changes.

Simplifying equations right from the start helps decrease complexity, making solving systems more straightforward. It aids in avoiding large numbers and cumbersome calculations in later steps, which can be a common issue in algebra.