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### 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!

## Check your understanding

# 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.

### 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.

## Check your understanding

# Challenge problems

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