# System of Linear Equations

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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.

Contents

### 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 x + cos 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 x.

 \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 $x_{2}$ and $x_{3}$.

$x_{1} = 10 - 2(1) + 5(2)$ $x_{1} = 10 - 2 + 10$ $x_{1} = 18$

### System of Linear Equations

The system of linear equations is a set of linear equations with $n$ variables $x_{1}, x_{2},x_{3}$, …,$x_{n}$.

$x_{1} + 5x_{2} + 2x_{3} = 6$
$3x_{1} + x_{2} - x_{3} = 4$
$4x_{1} + 2x_{2} + 2x_{3} = 6$

The solution set $c_{1}, c_{2}, c_{3}$, …, $c_{n}$ when replaced with variables $x_{1}, x_{2},x_{3}$, …, $x_{n}$

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

$x_{1} = 1, x_{2} = 1$ and $x_{3}= 0$

will satisfy all the equations.

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