Solving Systems with Matrices

Can’t figure out how to solve systems using Elimination Method? Substitution Method? Graphing? Why not try…

What is a system?

Systems are individual equations for lines. These equations usually start in Standard Form (ax + by = c), but not always. Most systems are linear equations (make a straight line), but not always. This lesson only addresses two equations given in Standard Form.

When we say “Solve a System”, we are looking for where the lines come together or cross. This is the solution to a system and is usually written as a point (x, y).

There are three possible answers to a system.

A point (x, y)No Solutions (lines are parallel and don’t touch)All Solutions (lines lie on top of each other)

This presentation does not address how to solve using any other method but matrices. Let’s start…

A matrix is a simple method for holding math objects in an organized pattern. They are written in two different ways. It doesn’t matter which visual you choose…

or

Let’s look at how they are used.

A question will look like this…

5x + 2y = 92x – 4y = 18

and

Let’s focus on the process, not the answer. So here’s the answer (3, -3). Now, the process…

Place your numbers into the matrix. They will go in two of them to make it easier.

5 22 -4

918 See. Easy and organized.

5x + 2y = 92x – 4y = 18

We are going to solve this system using a matrix. There will be three steps:

1. Find the Determinant

2. Find the X Value

3. Find the Y Value

Oh, How Easy!

5 22 -4

918

5x + 2y = 92x – 4y = 18First, the Determinant:

5 22 -4

918

Don’t Blink! This goes fast.(5 • -4) – (2 • 2)

Wow, that was easy! Do I have to do it again? I thought not.

Now Simplify: (5 • -4) – (2 • 2) = -20 – 4 = -24

The Determinant (D) is -24

5x + 2y = 92x – 4y = 18

D: -24

52

918

Now let’s find something called Dx…

2-4

First, get rid of the two x’s. Throw them away!

52Now replace them with the

Other matrix

918

To find Dx, get rid of the x and replace.

5x + 2y = 92x – 4y = 18Now finish finding Dx:

9 218 -4

Do it again!(9 • -4) – (18 • 2)

Keep it up…

Now Simplify: (9 • -4) – (18 • 2) = -36 – 36 = -72

Dx is -72

D: -24

5x + 2y = 92x – 4y = 18

D: -24 Dx: -72

52

918

Now for Dy…

2-4

Almost the same: get rid of the two y’s this time.

Now replace them with theOther matrix

918

To find Dy, get rid of the y and replace.

5x + 2y = 92x – 4y = 18Now finish finding Dy:

5 92 18

One more time!(5 • 18) – (2 • 9)

Now Simplify: (5 • 18) – (2 • 9) = 90 – 18 = 72

Dy is 72

D: -24 Dx: -72

5x + 2y = 92x – 4y = 18The final steps…

D: -24 Dx: -72 Dy: 72

Let’s go back on how to find x and y:

x = Dx y = Dy D D

x = -72 y = 72 -24 -24

Use your calculator if you have to…x = 3 y = -3

(3, -3)

3x + 5y = 116x – 2y = 10

I know it looks complicated at first. Let me show you on one page how easy it really is.

First, the problem…

Now create three matrices…

3 56 -2D = 11 5

10 -2Dx = 3 116 10Dy =

Now the math…

D(3• -2) – (6 • 5)

Look carefully where everything goes!

-36

Dx

(11• -2) – (10 • 5)-72

Dy(3• 10) – (6 • 11)

-36

Answer:x = Dx/D = -72/-36 = 2y = Dy/D = -36/-36 = 1

As you do your worksheet, please check your answers with the Excel Spreadsheet that has been shared with you. Plug in your values and check your own answers.