Systems of Linear Algebra with Graphs and Solution

1. “SYSTEMS OF LINEAR EQUATION”
MATHEMATICS
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2. INDEX
SYSTEMS OF LINEAR EQUATION
1. TWO VARIABLE SYSTEM OF EQUATION
• Systems of equation
• Graphical analysis
• Graphical solution
• The elimination procedure
2. GAUSSIAN ELIMINATION METHOD
• The general idea
• The method
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3. SYSTEMS OF LINEAR EQUATION
TWO VARIABLE SYSTEM OF EQUATION
 Systems of equation
• A system of equation is set consisting of more than one equation.
• One way to characterize a system of linear equation is by its dimensions.
 Dimensions
• If a system of equation consists of m equations and n variables, we say that this
system is an “m by n” system, or that it has dimensions m×n.
Example:
• A system of equations involving 2 equations and 2 variables is described as
having dimensions 2 X 2.
• A system consisting of 15 equations and 10 variables is said to be a (15 X 10)
system.
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4. Solution Set
• Using set notation, we would want to identify the solution set S, where
S = {(x,y)I5x + 10y = 20 and 3x + 4y = 10}
• The set of all solutions to an equation is called the solution set S to the
equation. The solution set S for a system of linear equations may be a
null set, a finite set, or an infinite set.
SOLUTION SET
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5. GRAPHICAL ANALYSIS
Solution to any linear equation system is the value of unknowns that satisfies all the
equation, graphically as shown in the figure its an intersection of two or more lines. System
of equations have three different types of solution sets that might exist:
• Unique Solution
• No Solution
• Infinitely Many Solutions
Unique solution No solution Infinitely many
solutions
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6. GRAPHICAL SOLUTION
Graphical solutions approaches are possible for two-variable systems of equations. To solve
a system of linear equations graphically we graph both equations in the same coordinate
system. The solution to the system will be in the point where the two lines intersect.
• Example:
Y=2x+2
y+=x-1
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7. THE ELIMINATION PROCEDURE
One popular method of solving two and three variable is the elimination method. To solve a
system of equations by elimination we transform the system such that one variable "cancels out".
This includes following steps:
• Given a system of equations, the two or multiplies of two equations, are added so as to
eliminate one of the two variables.
• The resultant equation is stated in terms of the remaining variables.
• This equation can be solved for the remaining variable.
Example:
Solve the system of equations by elimination:
3x−y=5
x+y=3
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8. EXAMPLE
Solution:
In this example we will cancel out the y term. To do so, we can add the equations together:
3x−y=5
x+y=3}
−−−−−−−−−−Add equations
4x=8
Now we can find: x=2
In order to solve for y, take the value for x and substitute it back into either one of the original equations
x+y=3
2+y=3
y=1
The solution is (x,y)=(2,1).
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9. GAUSSIAN ELIMINATION METHOD
THE GENERAL IDEA
•The Gaussian elimination method begins with the original system of equations and
transform it, using row operations, into an equivalent system from which the solution
may be read directly.
• An equivalent system is one which has the same solution set as the original system.
BASIC ROW OPERATIONS
I. Both sides of an equation may be multiplied by a non zero constant.
II. Non zero multiplies of one equation may be added to an other equation.
III. The order of equations may be interchanged.
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10. The method
•The general idea of Gaussian elimination method is to transform an original
system of equations into diagonal form by repeatedly applying the three basic
row operations.
•Example
2x+5y=10
3x-4y=-5
Would be written as
2 5 10
3 -4 -5
•The vertical line is used to the left and right sides of the equations.
•Column to the left of the vertical line contains all the coefficients for one of
the variables in the system
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