Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns...
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Transcript of Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns...
![Page 1: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/1.jpg)
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
![Page 2: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/2.jpg)
• 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
![Page 3: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/3.jpg)
• 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
![Page 4: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/4.jpg)
LINEAR SYSTEM
CONSISTENT INCONSISTENT
UNIQUE SOLUTION
INFINITELY MANY SOLUTION
NO SOLUTION
INDEPENDENT DEPENDENT
![Page 5: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/5.jpg)
l
m
ll
mm
A unique solution No Solution Infinitely many solutions
![Page 6: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/6.jpg)
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
![Page 7: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/7.jpg)
Infinite Solutions:
Non-Unique Solutions
• a pair of equations that have the same slope and y-intercept.
• also call a Dependent System
![Page 8: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/8.jpg)
Non-Unique Solutions
One Solution: • the lines of two equations intersect
• also called an Independent System
![Page 9: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/9.jpg)
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
![Page 10: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/10.jpg)
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.
![Page 11: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/11.jpg)
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
![Page 12: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/12.jpg)
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
![Page 13: Linear Equation The equation Which express the real or complex quantity b in terms of the unknowns and the real or complex constants is called a linear.](https://reader036.fdocuments.net/reader036/viewer/2022082819/56649f2a5503460f94c44ce9/html5/thumbnails/13.jpg)
• 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.