Show that the following system of equations has unique solutions. Find its solution by thereduction method𝑥 + 𝑦 + 𝑧 = 2, 𝑥 − 2𝑦 + 𝑧 = 8, 3𝑥 + 𝑦 + 𝑧 = 4

Sure, let's solve this system of equations step by step.

The system of equations is:

1. x + y + z = 2
2. x - 2y + z = 8
3. 3x + y + z = 4

Step 1: Subtract the first equation from the second and third equations to eliminate x:

Subtracting equation 1 from equation 2 gives: -3y = 6, or y = -2. Subtracting equation 1 from equation 3 gives: 2x = 2, or x = 1.

Step 2: Substitute x = 1 and y = -2 into the first equation to solve for z:

1 - 2 + z = 2, or z = 3.

So, the solution to the system of equations is x = 1, y = -2, z = 3.

This system of equations has a unique solution because each equation is linear and they are not multiples of each other, meaning they intersect at a single point.