Discuss the nature of solution of the following system of equation. x + 2y – z = 6; –3x – 2y + 5z = –12; x – 2z = 3 - Mathematics

Unveiling the Secrets of a System of Equations: A Step-by-Step Exploration!

Hey everyone! Ever stumbled upon a system of linear equations and wondered about the kind of solution it holds? Does it have one unique answer, no answer at all, or perhaps an infinite number of possibilities? Today, we're diving deep into one such system to uncover its secrets!

Let's consider the following system of equations:

1. x + 2y – z = 6                ... (Eq. 1)
2. –3x – 2y + 5z = –12      ... (Eq. 2)
3. x – 2z = 3                           ... (Eq. 3)

Our mission is to determine the "nature" of its solution. Let's get started!

Step 1: Combining Forces (Equations 1 and 2)

Let's try to simplify things by combining the first two equations. We can do this by adding Equation 1 and Equation 2:

  x+ 2y– z=6

+–3x– 2y+ 5z=–12

  –2x+ 0y+ 4z=–6

Notice how the +2y and –2y terms cancel each other out – neat, right?

This simplifies to:

–2x + 4z = –6                ... (Eq. 4)

Step 2: The Next Combination (Equations 3 modified and 4)

Now we have a new equation (Eq. 4) that only involves x and z. Let's bring in Equation 3, which also only involves x and z:

Equation 3:      x – 2z = 3
Equation 4: –2x + 4z = –6

To make these easier to combine, let's multiply Equation 3 by 2. This way, the coefficients of x (and z) will be opposites:

2 * (x – 2z) = 2 * (3)
2x – 4z = 6                ... (Eq. 3 modified)

Now, let's add this modified Equation 3 to Equation 4:

  2x– 4z=6

+–2x+ 4z=–6

  0x+ 0z=0

Look what happens!

0 = 0

Step 3: Interpreting the Grand Finale!

When our algebraic manipulations lead us to an identity like "0 = 0", it's a special signal! This tells us that the equations in the system are dependent on each other. Instead of pointing to a single, unique point of intersection (one solution) or being contradictory (no solution), they essentially describe overlapping scenarios.

Conclusion:

The result 0 = 0 indicates that the system of equations is consistent and has an infinite number of solutions. This means there isn't just one set of (x, y, z) values that satisfies all three equations, but countless combinations!

Isn't math fascinating? A few simple steps of algebra can reveal so much about the underlying structure of these equations! Stay tuned for more mathematical adventures!