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## Algebra (all content)

### Course: Algebra (all content)>Unit 20

Lesson 2: Representing linear systems of equations with augmented matrices

# Representing linear systems with matrices

Learn how systems of linear equations can be represented by augmented matrices.
A matrix is a rectangular arrangement of numbers into rows and columns.
Matrices can be used to solve systems of equations. But first, we must learn how to represent systems with matrices.

# Representing a linear system with matrices

A system of equations can be represented by an augmented matrix.
In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
In this way, we can see that augmented matrices are a shorthand way of writing systems of equations. The organization of the numbers into the matrix makes it unnecessary to write various symbols like $x$, $y$, and $=$ , yet all of the information is still there!

1) Which matrix represents the system?
$\begin{array}{rl}2x+3y& =8\\ 5x+2y& =2\end{array}$

2) Write the following system of equations as an augmented matrix.
$\begin{array}{rl}7x+4y& =3\\ 6x+3y& =5\end{array}$

# Let's look at another example

Now that we have the basics, let's take a look at a slightly more complicated example.

### Example

Write the following system of equations as an augmented matrix.
$\begin{array}{rl}3x-2y& =4\\ x+5z& =-3\\ -4x-y+3z& =0\end{array}$

### Solution

To make things easier, let's rewrite the system to show each of the coefficients clearly. If a variable term is not written in an equation, it means that the coefficient is $0$.
This corresponds to the following augmented matrix.
Again, notice how each column corresponds to a variable ($x$, $y$, $z$) or the $\text{constants}$. Also notice that the numbers in each row correspond to the coefficients in the same equation.
In general, before converting a system into an augmented matrix, be sure that the variables appear in the same order in each equation, and that the constant terms are isolated on one side.

3) Which matrix represents the system?
$\begin{array}{rl}3w-2x+y+5z& =10\\ w+2y-4z& =5\end{array}$

4) Write the following system of equations as an augmented matrix.
$\begin{array}{rl}-a+b-2c& =12\\ 3a+b& =-8\end{array}$

# Challenge problems

5*) Which system is represented by the augmented matrix?

6*) Which matrix represents the system?
$\begin{array}{rl}3x+2& =12y\\ -8y& =2x+15\end{array}$

## Want to join the conversation?

• Is there a reason the vertical line representing = is not used in this instance?
• Although I personally agree that it is very useful.
• Why does the + sign turn into - when put into standard form?
• Say you have the equation 3x+2y+8=0.
Then to get this equation in the standard form you will subtract both sides of the equation by 8. And you'll get 3x+2y=-8. So that's how you get the negative sign.
But that doesn't happen always. If you have 3x+2y-8=0, then you'll get 3x+2y=8.
Hope that helps. :)
• Do the matrix have to be in alphabetical order?
• When Sal writes letters in the matrices, they are just variables; they can be anything they want. So, no, they don't have to be in alphabetical order. It just makes it more simple to see and read.
• Why are system of equations represented in matrix format if it's harder to simplify? How do you solve a matrix system of equations?
• There are things called matrix row operations that let you rearrange the entries of a matrix while leaving the underlying system of equations "the same". Each row operation corresponds to a valid operation on the system of equations, like adding two equations together, writing them in a different order, or multiplying by a constant.

You can perform these matrix operations in a rote, mechanical way to find the solution set to any system of equations (if one exists). The goal of representing systems of equations like this is to remove the creative element from solving systems and provide an algorithm for solving them.
• So when there is a variable in one of the equations but not in the other the number 0 is like a placeholder correct?
• that's correct yeah the placeholder of the other equation when the variable is not there
(1 vote)
• What exactly would a 2x2 matrix represent? Would column 1 represent x and column 2 represent a constant x is = to? Would this simply represent 2 equations defining x?
• Typically, each row in a matrix represents an equation and each column represents a variable. A 2x2 matrix could be used to represent two linear equations. For example, we could put y = 3x + 5 and y = -x + 2 into the following matrix.

| 3 5 |
|-1 2 |
• How to find a determinant and an inverse of a given matrix
• The inverse of a matrix exists if and only if the determinant is nonzero. To find the inverse of a matrix, we write a new extended matrix with the identity on the right. Then we completely row reduce, the resulting matrix on the right will be the inverse matrix
(1 vote)
• Can we represent the augmented matrix of a set of equations with the variables jumbled up in any manner if the equations consist of (x,y,z) and constants
eg. 4x + 6y - 5z = 12
12x - 3y + 4z = 5
can we represent the augmented matrix as the following;
[4 -5 6 12]
[12 4 -3 5]
switching the (y) with (z) or should it be in the same order as it is given in the equation/ the alphabetical order??
• the equation itself can be jumbled up and has been in the final example on the page. However when putting it back into a matrix it needs to be x,y,z then the answer. in their respective rows.
• Why do numbers 5 and 6 have asterisks in the question? There isn't any little footnote or footer or whatever you want to call it there. Here is #5 as follows, 5*) Which system is represented by the augmented matrix?, and also #6 as follows, 6*) Which matrix represents the system?
• Those are the Challenge problems, so I believe the asterisks are used to indicate they are harder problems.
• In what context would it make more sense to use a matrix than to use the equations? Surely it's easier to see and understand numbers in equation form?