System of Four Linear Equations Solver

What is a System of Four Linear Equations?

Imagine you have four different puzzles, and each puzzle gives you a clue about four hidden numbers. A System of Four Linear Equations is like these four puzzles working together! It helps us find four special numbers that make all four puzzles true at the same time. It's like finding the perfect combination to unlock a treasure chest with four different locks!

How to Calculate Four-Variable Systems

To solve a four-variable system, we look at four equations all at once. We're trying to find values for four unknown numbers (usually called w, x, y, and z) that work for all four equations. It's like following four different treasure maps at the same time - the spot where all four maps point is our answer!

Formula

A system of four linear equations looks like this:

\[ a_1w + b_1x + c_1y + d_1z = e_1 \]

\[ a_2w + b_2x + c_2y + d_2z = e_2 \]

\[ a_3w + b_3x + c_3y + d_3z = e_3 \]

\[ a_4w + b_4x + c_4y + d_4z = e_4 \]

Here's what these letters mean:

• \(w\), \(x\), \(y\), and \(z\) are the unknown numbers we're trying to find
• \(a_1\), \(b_1\), \(c_1\), \(d_1\), and \(e_1\) are numbers in the first equation
• \(a_2\), \(b_2\), \(c_2\), \(d_2\), and \(e_2\) are numbers in the second equation
• \(a_3\), \(b_3\), \(c_3\), \(d_3\), and \(e_3\) are numbers in the third equation
• \(a_4\), \(b_4\), \(c_4\), \(d_4\), and \(e_4\) are numbers in the fourth equation

Calculation Steps

1. Write down all four equations
2. Use substitution or elimination to combine equations
3. Keep combining until you have one equation with one unknown
4. Solve for that unknown
5. Use the found value to solve for the other unknowns
6. Check if the values work in all four original equations

Example and Visual Representation

Let's solve this system:

\[ w + x + y + z = 10 \]

\[ 2w - x + y + z = 8 \]

\[ 3w + x - y + z = 10 \]

\[ 4w + x + y - z = 12 \]

Our solver finds that w = 1, x = 2, y = 3, and z = 4

Let's show this in a fun way:

In this picture, each colored circle represents one of our unknown numbers. The lines connecting them show how they all work together to solve our puzzle. It's like they're all holding hands to make our equations true!