Mathematics Supervision For The Experimental Language Schools
c
Prepared by: Nabil Z. Yanny March 2007


I ) The G. Cramer’s Rule 1750
If we have the system of n linear equation in n unknown






2 ) Two Equations in Two unknown


3 ) Equation of a plane







4) System of Two Equations in Three Unknown



Case 1: The Two Planes Intersect at a sraight line
When at least one of the following det.s is non zero
The two eq.s (1) and (2) has infinity
Solutions some unknown take arbitrary value and then get the others
Case 2: The Two Planes are parallel
When all det.s (3) are equal to zero, but one of the following det.s is non
zero


Then the two planes are parallel and the system has no solution

Case 3:The two planes are coincident
When all det.s in (3) and (4) are equal to zero , then the two equations reduces to
A single eq. And then the system has infinity of solution , some unknown take arbitrary value and then get the others
5) System of three equations in three unknown

I ) Unique Solution

EXC. Solve the following equations
3x + 4y + 2z = 5 , 5x – 6y – 4z = - 3 , - 4x + 5y + 3z = 1
SOLU.

II ) No Solution
( i ) If the coeff t. of the three planes are proportional , i.e. the three Planes are
parallels . then there is on solutions
EXC. Solve the following equations
X – 2y – z = 4 , 2X – 4y –2 z = 5 , 3X – 6y – 3z = 7
SOLU.

( ii ) If the coeff t. and the constant terms of two planes are proportional
i.e. two Planes are coincident and parallel to the third plane

EXC. Solve the following equations
7X – 4y + z = 5 , 14X – 8y + 2 z = 10 , 21X – 12y + 3z = 12


SOLU.
Then the first two planes are
Coincident and parallel to the third plane , i.e. the system reduces to two equations
Then there are no solutions
( iii ) If the coeff t. of the first two planes are proportional
i.e. the first two Planes are parallel , and the third plane cut them

EXC. Solve the following equations
X – 2y - z = 5 , X – 2y - z = 15 , 2X – 4y + 2z = 3
SOLU.

Then the first two planes are
Parallel and the third plane cut them, and then there are no solutions
( iv )
Hence all the three planes are parallel to a single straight line, and the three planes
Form a prismatic surface
EXC. Solve the following equations
x + y + z = 5 , x – y + z = 1 , x + z = 2
SOLU. the three planes are not parallel
Or coincides Add the two first equations we get
x + z = 3 which contradicts with The third eq. , then there is no solution
and the three plane form prismatic Surface
III ) Infinity of solutions
( i ) If the three planes are coincides ,
Then
And the three equations reduces to one equation
In three unknown, then two of them take
Arbitrary value, then get the third unknown
EXC. Solve the following equations
x - y + 2z = 3 , 2x – 2y + 4z = 6 , 3x – 3y + 6z = 9
SOLU. The three equations represent one plane
Then y = x + 2z – 3 , x and z take
Any value then get the value of y
( ii ) If the coefft. and the constant terms of the first two equations are proportional
i.e. the first two planes are coincides , i.e. represent one plane , and the third plane
cut it in a straight line , then

EXC. Solve the following equations
x - y + z =1 , 3x – 3y + 3z = 3 , x + y - 2z = 1
SOLU. The first two planes are coincides i.e. represent one plane, and the third
Plane cut it in a straight line, then
From equations of the first and third planes we get 2x – z = 2
Then z = 2x – 2 and y = 3x – 3 , if x take any value , then get value of y and z
( iii ) If
If one of the three equation is
a Consequence of the other two, then the
System Reduces to two equations in three
Unknown and has infinity of solutions
The three planes passing through an
Straight line called a pencil of planes
EXC. Solve the following equations
x + y + z =5 , x –y + z = 1 , x + z = 3
SOLU.
From equations of first and second planes we get x + z = 3
Then x = 3 – z and y = 2

Note: The triple scalar product
The triple scalar product

 

 

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