1. Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
Example : The equations
Following equations are not linear,
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2. Linear equations can have one or more variables
Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state. Linear equations do not include exponents.
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3. Figure
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4. Linear equation
A linear equation in unknowns is an equation that can be put in the standard form
Where and are constants. The constant is called the coefficient of and is called the constant term of the equation.
Example:
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5. Solutions
A solution of the linear equation is a list of values for the unknowns of numbers such that the equation is satisfied when we substitute The set of all solutions of the equation is called its solution set or sometimes the general solution of the equation.
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6. Systems of linear equations
An arbitrary system of linear equations in unknowns can be written as Where are the unknowns and the subscripted and denote constants. The system is called system. If the system is called square.
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7. The system (i) is said to be homogeneous if all the constant terms are zero. ie., if , otherwise the system is non-homogeneous. Homogeneous system I f at least one constant of the set is not zero then the system (i) is called a non-homogeneous .
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8. Example
Homogeneous Non-homogeneous
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9. Solution
If all are known called a particular solution. If all are not known called a general solution. is a solution of the system i.e, if it satisfies each equation of the system then this are called particular solution. The set of all particular solutions of the system is called a general solution.
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10. Solution(contd.)
Every homogeneous system of linear equation is consistent, since all such system have as solution. This solution is called the zero or trivial solution. If is a solution of the homogeneous system
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11. Solution(contd.)
If is a solution of the homogeneous system and if at least one is not zero, it is called a non –zero non-zero or non-trivial solution
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12. Echelon Form and Free Variables
If the disappearance of the leading variable is increased one line by another line is increased then the reduced system is called the echelon form Example:
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13. Echelon Form and Free Variables(contd.)
The variables which do not appear at the beginning are called free variables Example: Numbers of free variables and what are they: In the above example there are 3 equations with 4 variables So 4-3=1 free variables. And leading variable is missing so is free variable.
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14. Consistent & Inconsistent
A system of linear equations is said to be consistent if no linear combination of its equations is of the form Otherwise the system is inconsistent.
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15. At a Glance Linear equation Solution System of linear equation
Solution of the System of linear equations
Homogeneous and non-homogeneous System of linear equations
Solution of Homogeneous and non-homogeneous System of linear equations
Echelon form
Free variables
How many free variables and what are they?
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16. System of Linear Equation
Non-homogeneous system
of linear equation
Homogeneous system
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17. Non-homogeneous Linear Systems
System of linear Equations
Consistent
Unique Solution
In echelon form free variables does not exist
More than one solution
In echelon form free variables exist
Inconsistent
No solution
In echelon form at least one equation will appear of the form
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18. Problems
Let the system Solution: The given system is
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19. So from Again from Finally The solution Is
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20. Similar problem: Solve Solution: The given system is
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21. Similar problem: Find the solution of the following system of linear equation by reducing it to the echelon form (i) (ii) no solution
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22. Augmented Matrices
If we mentally keep track of the location of the the and the a system of linear equations in unknowns can be Abbreviated by writing only the rectangular array of members This is called the augmented matrices for the system.
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23. Example
The augmented matrices for the above system of equation
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24. Problem(Augmented Matrices)
Solve: The augmented matrices for the above system of equation is
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25. . ~
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26. Solve the following system Solution: The given system is
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27. The above system (iii) shows that it has two equations of four Unknowns, so the system has more than one solution. Here the Number of free variables 4-2=2. And these are Let and We get from Thus we have the solution
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28. Solve the following system Solution: The given system is
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29. ~
Here 4-3=1 free variable which is , Let
We get
The solution is
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30. Solve the following system Solution: The given system is
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31. ~
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32. ~
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33. ~
The solution is
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34. Problem: Show that the system has (1) a unique solution if (III) more than one solution if (III) no solution if Solution: The given system is
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35. Solution: The given system is
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36. Discussion: Case -1: The system (iv) is echelon form
Discussion: Case -1: The system (iv) is echelon form. It has a unique solution if the coefficient of z of the third equation is not zero ie if Case-II: It has more than one solution if . Since under this condition there exists in (iv) two equations in three unknowns. Case-III: then the system has no solution. Since under this condition the third equation of (iv) becomes the impossible solution.
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37. Problem: What relation may exist among the constants such that the following system has a solution Solution: The given system is
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38. (iii) & (i) are equivalent
(iii) & (i) are equivalent. Now for having a solution the system (iii) must be exist and it is possible if Similar problem: Determine the relationship among the constants under which the following system has a solution
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39. Problem: For what values of and the following system of linear equations has (i) no solution (ii) more than one solution (iii) a unique solution. Solution: The given system is
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40. Case-1: If and then the last equation of the system (iii) a non-zero real number, which is impossible .In This case the system has no solution. Case-II: If and then there are two equations in three unknowns. Since in this case the last equation of this system becomes
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41. Case-1II: If then system has three equations in the three unknowns
Case-1II: If then system has three equations in the three unknowns. In this case the system has a unique solution. Similar problem: e
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42. Problem: For what values of the following system of linear equations has (i) a unique solution (ii) more than one solution (iii) no solution. Solution: The given system is
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43. Solution: The given system is Case-I: Case-II: Case-III:
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44. Homogeneous Linear Systems
System of homogeneous linear Equations
Consistent
Zero Solution
In echelon form free variables does not exist
Non-zero solution
In echelon form free variables exist
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45. Problems (Homogeneous)
2. Solve
Solution: Given, ~
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46. ~ The system (iii) is echelon form in which equation with 5
variables. So there are 5-2=3 free variables and they are
Let
We have from
Thus the solution is
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47. Problems (Homogeneous)
Solve:
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48. Solve:
The given system is ~
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49. ~ The given system is Let
The system (iii) is echelon form in which equation with 4 variables. So there are 4-2= free variables and they are
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