Gaussian elimination

Imagine you have a puzzle with different colored blocks that you need to sort in a certain order. You have a set of simple rules that tell you how to move the blocks around, like "you can switch the places of two blocks" or "you can add or subtract a certain amount of blocks from each other."

Gaussian elimination is like using those simple rules to figure out how to solve a bigger puzzle, but instead of blocks, you're dealing with numbers. It's a fancy math tool that helps you solve systems of equations (which are basically math puzzles with a lot of rules) by breaking them down into smaller, easier puzzles.

Here's how it works:

Let's say you have this system of equations:

3x + 2y - z = 1

2x - 2y + 4z = -2

-x + 1/2y - z = 0

You want to solve for x, y, and z - in other words, figure out what numbers x, y, and z need to be in order for all three equations to be true at the same time.

First, you can use the simple rules to get rid of one of the variables. In this case, let's get rid of x. We can do that by using the first equation to "isolate" x, which means getting x by itself on one side of the equation:

3x + 2y - z = 1

3x = 1 - 2y + z

x = (1 - 2y + z)/3

Now we can substitute this expression for x into the other two equations, effectively getting rid of x altogether. This gives us two new equations:

2(-2y + 4z)/3 - (1 - 2y + z)/3 = -2

-(1 - 2y + z)/3 + 1/2y - z = 0

Simplify these equations a bit and you get:

-4y + 8z - 1 + 2y - z = -6

-3y + 6z - 2 + 3z = 0

Now you can use the same strategy to isolate another variable (let's say y), and substitute it into the remaining equation to get an expression for z. Finally, you can substitute the values you've found for y and z back into any of the original equations to get x.

This process of breaking down the system of equations into smaller, easier puzzles is called Gaussian elimination. It's kind of like the "divide and conquer" strategy you might use to solve a big problem by breaking it up into smaller pieces. Gaussian elimination is a powerful tool that can be used in all sorts of real-life situations, like analyzing data or solving engineering problems. But at its core, it's just a fancy way of using simple rules to solve big puzzles.