3D Linear Equation Solver: Calculate Solutions for Three-Variable Systems

What is a 3D Linear Equation Solver?

A 3D Linear Equation Solver is like a magical box that helps us find where three planes meet in space! It's a tool that finds values for three unknown numbers (usually called x, y, and z) that make three math sentences true at the same time. Imagine you're trying to find a special point where three giant sheets of paper intersect in a big room - that's what this solver does!

How to Calculate Three-Variable Systems

To solve a three-variable system, we look at three equations together. We find the values of x, y, and z that work for all three equations. It's like following three different treasure maps at once - the spot where all three maps point is our answer!

Formula

A system of three linear equations looks like this:

\[ a_1x + b_1y + c_1z = d_1 \]

\[ a_2x + b_2y + c_2z = d_2 \]

\[ a_3x + b_3y + c_3z = d_3 \]

Here's what these letters mean:

• \(x\), \(y\), and \(z\) are the unknown numbers we're trying to find
• \(a_1\), \(b_1\), \(c_1\), and \(d_1\) are numbers in the first equation
• \(a_2\), \(b_2\), \(c_2\), and \(d_2\) are numbers in the second equation
• \(a_3\), \(b_3\), \(c_3\), and \(d_3\) are numbers in the third equation

Calculation Steps

1. Write down all three equations
2. Use substitution or elimination to combine two equations
3. Solve the resulting two-variable system
4. Use the found values to solve for the third variable
5. Check if the values work in all three original equations

Example and Visual Representation

Let's solve this system:

\[ x + y + z = 6 \]

\[ 2x - y + z = 3 \]

\[ x + 2y - z = 3 \]

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

Let's show this in a 3D space:

In this 3D graph, each colored line represents where two of the planes intersect. The purple dot shows where all three planes meet, which is our solution (2, 1, 3).