Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Linear Equations and Matrices

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
• x1,x2, . . . ,xn are n unknowns.
• The aij and bi are the the given constants

A solution to the linear system (1) is a sequence of n numbers s1, s2, . . . , sn such that (1) is satisfied when x1 =
s1,x2 = s2, . . . ,xn = sn.

Example 1. Determine if the following systems are linear and check if the given s1, s2, . . . 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 b1 = b2 = · · · = bm = 0, we say that (1) is a homogeneous system. Note that a homogeneous system
always admits the solution x1 = x2 = · · · = xm = 0, called the trivial solution. A solution to a homogeneous
system where not all of x1,x2, . . . ,xm 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.