Matrix Solutions to Systems of Equations (3 Equations and 3 Unknowns)

An Interactive Applet powered by Sage and MathJax.

(By Prof. Gregory V. Bard)

Overview

On this page, we're going to explore the solution of a system of equations. In particular, we're going to look at a system that happens to have 3 equations and 3 unknowns---the variables x, y, and z. The reason for this choice is that it allows us to visualize what is happening as a 3-dimensional graph.

Whenever we have a system of linear equations, we should always think about using matrices. Matrix techniques are ideal for this kind of situation. The number of rows in the matrix is the number of equations, and the number of columns is one more than the number of variables. That's because each variable gets its own column, and the last column is intended for the constants on the other side of the equal sign. Since we have 3 equations and 3 unknowns, we expect then to have 3 rows and 4 columns.

Any system of linear equations has three possibilities, either it has...

• A unique solution.
• No solutions.
• Infinitely many solutions.

Part One: Exploring the Question Numerically.

Naturally, we're interested in finding the solution, but we need to know what the solution will look like. When we say "a unique solution" what we mean is that there will be a numerical value for each variable. In our case, that means x, y, and z, will each have a numerical value. Furthermore, when we plug in those numerical values, all three equations are simultaneously satisfied.

Note: When you drag the slider back and forth, the applet won't recompute the answers until you let go of the mouse button. There's also a 2-3 second delay, depending on the speed of your internet connection.

Part Two: Exploring the Question Graphically

blah blah

Just as equations involving x and y will produce graphs (such as lines and curves) in a two-dimensional space (such as on a piece of paper), equations involving x, y, and z will produce graphs (such as surfaces) floating in ordinary 3-dimensional space. You might or might not have been told that an equation of the form $2x + y + z =1$ has, as its graph, a plane floating in space. Therefore, each of our three equations represents a plane.

This graph has several pieces to it:

• The plane of the first equation is drawn in red.
• The plane of the second equation is drawn in yellow.
• The plane of the third equation is drawn in blue.
• The x-axis, y-axis, and z-axis, are lines drawn in apricot. (Hint: axis and apricot both start with the letter "a.")
• Next, the giant blah-colored sphere or round dot where the three spheres intersect represents the solution itself. It should be the case that the solution dot resides on each and every one of the three given planes.
• Any time two distinct planes meet in space, their overlap forms a line. That overlap is visible, in each case, as a white line. That's not important now, but it will be important in a later interact that shows the "no solution" case. Still, it is nice to note that the solution lies on each of the lines as well.

I don't know about you, but graphically, I was blah blah. After that, I had to blah blah. The graph was useful for me, but this is an excellent example of how a graph is only a picture---it is not enough, alone, to answer the big questions.

Part Three: blah blah

blah blah

Part Four: Closing Questions

Now here are some questions that you should be able to answer based on your work above.

1. blah
2. blah
3. There are no right or wrong answers for this one: How useful was the graph in helping you understand what is going on while solving this system of equations with a matrix?