**Definition 1**. A **system of m linear equations in n unknowns** is a
set of m linear equations each in n unknowns of

the form:

where

• x_{1},x_{2}, . . . ,x_{n} are n unknowns.

• The a_{ij} and b_{i} are the the given constants

A solution to the linear system (**1**) is a sequence of n numbers s_{1},
s_{2}, . . . , s_{n} such that (**1**) is satisfied when x_{1}
=

s_{1},x_{2} = s_{2}, . . . ,x_{n} = s_{n}.

**Example 1**. Determine if the following systems are linear and check if
the given s_{1}, s_{2}, . . . are their solutions

**Definition 2.**

• We say that the linear system We say that two linear systems are equivalent
if they both have exactly the same

solutions. is consistent if it has a solution. Otherwise it is called
inconsistent.

• When b_{1} = b_{2} = · · · = b_{m} = 0, we say that
(**1**) is a homogeneous system. Note that a homogeneous system

always admits the solution x_{1} = x_{2} = · · · = x_{m}
= 0, called the trivial solution. A solution to a homogeneous

system where not all of x_{1},x_{2}, . . . ,x_{m} are
zero is called a nontrivial solution.

**Equivalent linear systems and properties**. We say that two linear
systems are equivalent if they both have exactly

the same solutions.

Properties.

The system (**1**) is equivalent to a new linear system manipulated by:

1. Interchanging the ith and the jth equations

2. Multiplying an equation by a nonzero constant

3. Replacing ith equation by c times the jth equation plus the ith equation, i≠j.
That is, replacing

**Example 2. **Consider the system

with the solution x = 4,y = 2. Check that the following systems also have the same solution

Note: (**2b**)_{2}(**2a**)_{2}−3(**2a**)_{1}

Note: (2c)_{2}(-1/10)(2b)_{2}

**Solving linear systems - Method of elimination**. To
find a solution to a linear system, we use the method of

elimination; that is, we eliminate some variables by using the properties 1,2,3
above to get to simpler linear systems,

easier to solve.

**Example **3. 2, 6/p8, 16/p9.