Linear EquationThe equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear equation.

,..............2211 dxaxaxa nn

nxxx ...,,.........2,1

naaa ....,,.........2,1

• Consistent: If the Linear system has a solution, then it is called as Consistent.

• Inconsistent: If the linear system has no solution then it is said to be Inconsistent.

• Homogeneous Equation: If Then it is called a Homogeneous system.

• Trivial Solution: If Then it is called as the trivial solution.

0..........321 mbbbb

0............321 nxxxx

• Non-Trivial Solution: A solution to a homogeneous system in which not all of

are zero is called a non-trivial solution .

• Equivalent: If two systems of linear equations have exactly the same solution then it is called as they are equivalent.

nxxx .,,........., 21

LINEAR SYSTEM

CONSISTENT INCONSISTENT

UNIQUE SOLUTION

INFINITELY MANY SOLUTION

NO SOLUTION

INDEPENDENT DEPENDENT

l

m

ll

mm

A unique solution No Solution Infinitely many solutions

Non-Unique Solutions

No Solution: • when lines of a graph are parallel

• also called an Inconsistent System

• since they do not intersect, there is no solution

Infinite Solutions:

Non-Unique Solutions

• a pair of equations that have the same slope and y-intercept.

• also call a Dependent System

Non-Unique Solutions

One Solution: • the lines of two equations intersect

• also called an Independent System

Examples…

1)

Determine whether the following have one, none, or infinite solutions by looking at the slope and y-intercepts

2y + x = 8

y = 2x + 4

3) 2) x - 5y = 10

-5y = -x +6

y = -6x + 8

y + 6x = 8

ANS: One Solution

ANS: No Solution

ANS: Infinite Solutions

Elimination MethodExamine our original system:

-2x + y = 4 -6x + y = 0

Notice that the y coefficients are 1, therefore we can multiply either equation by -1 and add the system, thus eliminating the y variable.

Question 1Solve the system using elimination method:

2x + 5y = 73x + y = -9

The solution is:a. (12, -4)b. (-4, 12)c. (4, -21)d. No Solution

Question 2

-2

-1

0

1

2

3

4

5

6

-1 0 1 2

How many solutions exist for the system at the right?

a. 0b. 1c. 2d. Infinite

• Example 1• The sum of two numbers is 37. One number is 5 larger than the other. What are the

numbers?• Step 1: Let x = smaller number.       y = larger number.

Step 2:  There are two relationships.     (a) The Sum is 37.        x+y=37     (b) One is 5 larger than the other.        y=x+5        { x+y=37      y=x+5 (1) (2)

Step 3:   { x+y=37      y=x+5 (1) (2)  We can easily solve this system by substitution. Substitute x+5 for y in equation 1.      x+(x+5) = 37 x+x+5 = 37 2x+5 = 37 2x = 32 x = 16 Then, y = 16+5 = 21. x = 16, y = 21

Step 4:  The Sum is 37.      x+y = 37 16+21 = 37 Is this correct? 37 = 37 Yes, this is correct.  One is 5 larger than the other.      y = x+5 21 = 16+5 Is this correct? 21 = 21 Yes, this is correct.

Step 5:  The two numbers are 16 and 21.