The document discusses normal equations, which are formed by multiplying each observation equation by the coefficient of the unknown quantity and adding the results. This allows the most probable values of the unknowns to be determined. The document provides an example of forming normal equations for three unknowns, x, y, and z, from three observation equations of equal and unequal weights. The normal equations are presented as the solution for determining the values of x, y, and z.

1. Computation of indirectly observed
quantities- Normal equations

2. NORMAL EQUATIONS
• A normal equation is the one formed by multiplying each equation by
the coefficient of the unknown whose normal equation is to be found
and by adding the equation thus formed.
• As the number of normal equations is the same as the number of
unknowns, the most probable values of the unknowns can be found
from the equations.

3. Contd…

4. Contd…

5. Rules for Normal Equations
For equations of equal weights
• To form normal equation for each of the unknown quantity,
multiply each observation equation by the algebraic
coefficient of that unknown quantity in that equation and add
the result
For equations of different weights
• To form normal equation for each of the unknown quantity,
multiply each observation equation with the product of the
algebraic coefficient of that unknown quantity in that equation
and the weight of that observation and add the result.
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6. Qn.1.(a) Form the normal equations for x, y and z in the following
equations of equal weight.
3x + 3y + z -4 = 0 …..(1)
x+2y+2z- 6 = 0 …..(2)
5x +y+4z – 21= 0 …..(3)
(b) If the weights of the above equations are 2,3 and 1 respectively, form
the normal equations for x, y and z.

7. Solution.
(a) The normal equation of an unknown quantity is formed by
multiplying each equation by the algebraic coefficient of that
unknown quantity in that equation and adding the result.
9x + 9y + 3z - 12 = 0
x + 2y + 2z - 6 = 0
25x + 5y + 20z - 105 =.0
Normal equation for x is 35x + 16y + 25z -123 =0

8. 9x + 9y + 3z – 12 =0
2x + 4y + 4z – 12 =0
5x + y + 4z – 21 = 0
Normal equation for y is 16 x +14 y +11 z – 45 = 0
3x + 3y + z – 4 = 0
2x + 4y + 4z – 12 = 0
20x + 4y + 16z – 84 = 0
Normal equation for z is 25 x +11 y +21 z – 100 = 0

9. Contd…
• Hence the normal equations for x, y and z are
35 x +16 y + 25 z – 123=0
16 x +4 y + 11 z – 45=0
25 x +11y + 21z – 100=0
(b) In equations (1), (2) and (3) the products of coefficients of x and
weight of respective equations are : (3 * 2), (1 * 3) and (5 * 1).Hence
18 x +18 y + 6 z – 24=0 (from 1)
3 x + 6 y + 6 z – 18=0 (from 2)
25 x + 5 y +20 z –105=0 (from 3)
Normal equation for x is 46 x +29 y + 32 z -147 = 0
W = 2
W=3
W=1

10. Contd…
• Similarly, the product of coefficient of y and weight of each
equation, in the original equations are ( 3*2), (2 * 3) and (1 * 1)
respectively. Hence,
18 x +18 y + 6 z – 24= 0 (from 1)
6 x +12 y + 12 z – 36= 0 (from 2)
5 x + y + 4 z – 21= 0 (from 3)
Therefore, Normal equation for y is 29 x +31 y +22 z -81 =0
W = 2
W=3
W=1

11. Contd…
And , the product of coefficient of z and weight of each equation, in the
original equation are (1 * 2), (2 * 3) and (4 * 1) respectively. Hence ,
6 x +6 y + 2 z – 8=0 (from 1)
6 x +12 y + 12 z – 36=0 (from 2)
20 x +4 y + 16 z – 84=0 (from 3)
Normal equation for z is 32 x +22 y +30 z -128 =0
W = 2
W=3
W=1

12. Contd…
Hence the normal equations for x, y and z are as
follows:
46 x + 29 y + 32 z- 147 = 0
29 x +31 y +22 z - 81 = 0
32 x +22 y +30 z- 128 = 0