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### Course: Precalculus (Eureka Math/EngageNY)>Unit 2

Lesson 2: Topic C: Systems of linear equations

# 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.
(1 vote)
• 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 |
• the last one dont make any sense
• If you mean the last question from challenge problems, it formatted in a way to kinda trick you.
​3x+2=12y, which is not in the standard form. We can rewrite it as 3x-12y=-2. The second one is -8y= 2x+15, which is also not in standard form. Rewriting it we get 2x+8y=-15. Now if we write express this as a matrix, we get
[3 -12 -2 ]
[2 8 -15]
But there is no such option. So, lets revisit an equation and modify it so that it would match with one in the options. Take 2x+8y=-15 . Multiplying both sides by -1, we get -2x-8y=15. Now, we get
[ 3 -12 -2 ]
[-2 -8 15]
(Note: I didn't take the first equation since all the options have 3 in its first column, first row).
This is option C. Hence it is the answer.
• 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??