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 = 9 2x – 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 See. Easy and organized.

5x + 2y = 9 2x – 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!

5x + 2y = 9 2x – 4y = 18 First, the Determinant: 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 = 9 2x – 4y = 18 D: Now let’s find something called Dx… 2 -4 First, get rid of the two x’s. Throw them away! 5252 Now replace them with the Other matrix 9 18 To find Dx, get rid of the x and replace.

5x + 2y = 9 2x – 4y = 18 Now finish finding Dx: 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 = 9 2x – 4y = 18 D: -24 Dx: Now for Dy… 2 -4 Almost the same: get rid of the two y’s this time. Now replace them with the Other matrix 9 18 To find Dy, get rid of the y and replace.

5x + 2y = 9 2x – 4y = 18 Now finish finding Dy: 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 = 9 2x – 4y = 18 The final steps… D: -24 Dx: -72 Dy: 72 Let’s go back on how to find x and y: x = Dx y = Dy D x = -72 y = Use your calculator if you have to… x = 3 y = -3 (3, -3)

3x + 5y = 11 6x – 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… D = Dx = Dy = 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 = 2 y = 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.