Before you we try to understand the system of linear equations, we must understand the basic terminologies that we are going to use in the linear algebra.

### Linear Equations

Linear equation represents a straight line on a xy-plane or co-ordinate plane.

a_{1}x + a_{2}y = b

- The are
**real number constants**and both cannot be zero at the same time. - The and are called
**variables**. Hence, the equation is linear equation in variable and .

Therefore, linear equation in n variable is given as

a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + \dots + a_{n}x_{n} = b

where and are real number constants.

**Examples of linear equations**

Here are some examples of linear equations.

\begin{aligned} 3x + 5 = 13\\ \\ \frac{3}{4}y + 3z = 1 \end{aligned}

From the above examples, note that linear equations show following characteristics.

- There is not term in the equation with root or product.
- All variables are in power of 1.
- They do not appear as an argument of another function.

Here are examples of non-linear equations to help you understand the basic differences.

\begin{aligned} &2 + \sqrt{y} = 7\\ &x + 2yz + z = 6\\ &sin \hspace{3px} x + cos \hspace{3px}x = 1 \end{aligned}

### Solution Of Linear Equations

A linear equation

a_{1}x_{1} + a_{2}x_{2} + ... + a_{n}x_{n}

has a sequence of numbers

c_{1},c_{2},..., c_{n}

as **solution **which satisfy the equation once we substitute them with the variables.

For example,

a_{1}x_{1} + a_{2}x_{2} + a_{2}x_{2} + ... + a_{n}x_{n]} = b\\ \\

The solution is

Then,

will satisfy the equation.

The set of all solutions of a linear equation is called** solution set or general solution** to the linear equation.

#### How to find solution to linear equations ?

Now we will solve two linear equation using substitute method.

**Problem #1** : solve .

**Solution:**

We can solve like ordinary algebraic equation. Solve for x for solve for y.

Add to both side of the equation

x + 2y - 2y = 4 - 2y

now we have

x = 4 - 2y

Let’s give an arbitrary value to y. This way we get value for .

\begin{aligned} &y = 1\\ &x = 4 - 2y\\ &x = 4 - 2\\ &Therefore, \hspace{2mm}x = 2 \end{aligned}

Problem #2 : solve

——————————————————————————-

Solution:

————–

This time we can solve for two variables and get the third. We get following equations.

Let us give arbitrary value to and .

### System of Linear Equations

The **system of linear equations** is a set of linear equations with variables .

\begin{aligned} &x_{1} + 5x_{2} + 2x_{3} = 6\\ &3x_{1} + x_{2} - x_{3} = 4\\ &4x_{1} + 2x_{2} + 2x_{3} = 6 \end{aligned}

The solution set when replaced with variables

will satisfy every equation in the **linear system.** In the above system of linear equations,

and

will satisfy all the equations.

In the next post, we will discuss about **inconsistent **and **consistent **system of linear equations.