System of Linear Equations

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Unit 6

• System of Linear Equations

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6.1 Crammers rule of Solving System of Linear
Equations
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6.1 Crammers rule of Solving System of Linear
Equations

• Crammers Rule

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6.1 Crammers rule of Solving System of Linear
Equations
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P.202 Ex.6A
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6.2 Gaussian Elimination
A system of linear equations is said to be in
Echelon form if it is of the form
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6.2 Gaussian Elimination
A system of linear equations is said to be in
Echelon form if it is of the form
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6.2 Gaussian Elimination
e.g.
The process of reducing a given system of linear
equations in Echelon form is called Gaussian
Elimination.
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6.2 Gaussian Elimination
coefficient matrix
augmented matrix
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6.2 Gaussian Elimination
Elementary row operation on matrix A is any one
of the following three operations

• interchange two rows,
• multiply a row of A by a non-zero scalar,
• add a multiply of one row to another row.

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6.2 Gaussian Elimination
A matrix satisfying the following two conditions
is said to be in Echelon form

• The elements in the first k rows are not all
  zero those in other rows are all zero.
• The first non-zero element in each non-zero row
  is 1, and its appears in a column to the right of
  the first non-zero element of any preceding row.

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6.2 Gaussian Elimination
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6.2 Gaussian Elimination
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6.2 Gaussian Elimination
z t, y 2t 2, x -1 7t
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6.2 Gaussian Elimination
z -1, y 1, x 2
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6.2 Gaussian Elimination
w t z s, y 1 2s 2t, x -s 2t
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6.2 Gaussian Elimination
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P.209 Ex.6B
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6.3 Theory of system of Linear Equations
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6.3 Theory of system of Linear Equations
The system has a unique solution if and only if
the determinant of the coefficients ? 0.
If ? 0 and not all ?x, ?y and ?z are equal to
zero, then the system does not have solution.
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6.3 Theory of system of Linear Equations

• A system of equations (not necessarily
  linear) is said to be
• solvable or consistent if and only if it has a
  solution (the solution is not necessarily
  unique).
• unsolvable or inconsistent if and only if it has
  no solution.

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6.3 Theory of system of Linear Equations
A system of linear equations is said to be
homogeneous if and only if all the constant terms
equal zero.
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6.3 Theory of system of Linear Equations
If x y z 0, such solution is called a
trivial solution. Any solution with x, y and z
not all zero, of the system of equations is
called a non-trivial solution.
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P.217 Ex.6C
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6.4 the Method of Cross Multiplication
method of cross multiplication
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6.4 A Few Applications
(A) Area of a triangle with vertices (x1,
y1), (x2, y2) and (x3, x3)
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6.4 A Few Applications
(A) Area of a triangle with vertices (x1,
y1), (x2, y2) and (x3, x3)
Three distinct points A(x1, y1), B(x2, y2) and
C(x3, y3) are collinear if and only if
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6.4 A Few Applications
(B) Intersection of two straight lines
(1) unique solution (intersect at a point)
(2) no solution (no intersection)
They are parallel.
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6.4 A Few Applications
(B) Intersection of two straight lines
(3) no solution (just one straight line)
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6.4 A Few Applications
(C) Concurrency of three straight lines
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6.4 A Few Applications
(D) System of linear equation with two unknowns
but three equations
If the system of linear equations
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6.4 A Few Applications
(D) System of linear equation with two unknowns
but three equations
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P.233 Ex.6E