,,

: a IV5

:

. . , nocu , . . , . . . , N, Z, Q, R, , . (). . , . . . . (), . . , . , . 3 ey : 1. - , . : . 2. - , . , 1 2 = b, b . 3.

b = b a , 0 a = 0 b 0 a = 0 b = 0, .

2

. . , .

: . 1) 3 5(2 3) = -2 3 10 + 15 = -2 ( 1) -10 = -2 18 ( 2) . -10 = -20 ( 1) 20 = 10 ( 3) =2 , . 1 2 + 8 = 2 (6 + 4) 2 + 8 = 3 + 2 ( 1) . 2 2 = 3 8 ( 2) 0 = -5 ( 1) 2 9 7(2 ) + 10 = 3 (21 + 2 ) 14 7 + 10 = 14 + 3 -7 - 10 3 = 14 14 0 = 03

. 2) , . . 3)

.

, ax + by = c . y , , b c . . x y (0 Y0) x y . . x y, + by = c . 7 23 (8,2), (9,3), ( 5 ,- 5 ) (1). , , (,-6) . , x = 7, y =-3 (7,-3) (2). 2 5 2 5 x=a, y = - 3 a + 3 , (,- 3 + 3 ) . 2 5 (2) 2x + 3y = 5 y = 3 x + 3

(1) x y -6 = 0

x y a1x + b1y = c1 , a2x + b2y = c2 , . x y (X0,Y0) x X0 y Y0 . , . . .1) , , . .2) , . .3) , , .4

.4) , . x = m y = n. : x 2y = 3 2x + y = 11 x = 2y + 3 .2 2x + y = 11 x = 2y + 3 2(2y + 3) + y = 11 .3 x = 2y + 3 4y + 6 + y = 11 x = 2y + 3 5y = 11 6 x = 2y + 3 5 y= 5 =1 x = 5, y = 1 x 2y = 3 2x + y = 11 5 21 = 3 25 + 1 = 11 52 =3 10 + 1 = 11 3=3 11 = 11 + .4 5x + 2y = 1 x 2y = 5 6x = 6 x 2y = 5 6 x= 6 =1 1 2y = 5 x=1 -2y = 5 1 x=1 4 y= 2 x=1 y = -2 x=1 R(1,-2)

.3

.1

x = 21 + 3 = 5 y=1 x=5 y=1 R(5,1)

, . 1.

5

0100090000037800000002001c00000000000400000003010800050000000b0200000000050000000c 026e046705040000002e0118001c000000fb021000070000000000bc02000000ee01020222537973746 56d0004670500002c820000d03c110004ee8339289da6020c020000040000002d01000004000000020 101001c000000fb029cff0000000000009001000000ee0440001254696d6573204e657720526f6d616e0 000000000000000000000000000000000040000002d010100050000000902000000020d000000320a5 a000000010004000000000065056b04204d2d00040000002d010000030000000000 1 2

6

0100090000037800000002001c00000000000400000003010800050000000b0200000000050000000c 026e046705040000002e0118001c000000fb021000070000000000bc02000000ee0102022253797374 656d0004670500002c820000d03c110004ee8339289da6020c020000040000002d0100000400000002 0101001c000000fb029cff0000000000009001000000ee0440001254696d6573204e657720526f6d616 e0000000000000000000000000000000000040000002d010100050000000902000000020d000000320 a5a000000010004000000000065056b04204d2d00040000002d010000030000000000 3

.1 a1x + b1y = c1 a2 x + b2y = c2 (X0,Y0), .

.2 , , ( ). .3 ( ).

2. e , a , : 3x 2y = 6 x+y =7/2 3x 2 y = 6 2 x + 2 y = 14 + . 2 5x = 20 3x 2y = 6 x=4 3x 2y = 6 x=4 34 2y = 6 x=4 12 2y = 6 x = 4, y = 3 R(4,3) .4 .2 .3 .2

3. 7

, , . . m n x1, x2,...xn : a11 x1 + a12 x2 + ...+ a1n xn = b1 a21 x1 + a22 x2 + ...+ a2n xn = b2 . (1) . . am1 x1 + am2 x2 + ...+ amn xn = bm aij( i = 1,2,...,m; j = 1,2,...,n) bi( i = 1,2,...,m) ( ). ij , bij . bi = 0 ( i = 1,2,...m) . , (1) , . . n (1,...,n) X1 = C1,...,Xn = Cn (1)

aj =1

n

ij j

c = bi (i = 1,...,m)

(1) Rn ( n) . , (1) , = . , . , (1) , . (0,0,0,...,0). , , . . . , . , . 1. R 3x1 x2 = -98

5x1 + 4x2 = 2 x1 = -2, x2 = 3 . (-2,3). 2x1 + x2 = -1 -x1 + x2 = 5 x1 + 2 x2 = 4 , , (-2,3). , . : 1) i- j- 2) M i- 0 R 3) i- j- 1. . . Carl Friedrich Gauss (1777-1855) , . . . . 18. . a11x1 + a12x2 + ... + a1nxn = b1 a21x1 + a22x2 + ... + a2nxn = b2 . . . am1x1 + am2x2 + ... + amnxn = bm

(1)

m n. a11 0. a31 a 21 a11 . a11 . , (1). a11x1 + a12x2 +(1)

a13x3 + ... + a1nxn = b(1) (1) (1) (1) (1) (1)

22 (2)

x 2 + 23 x3 + ...+ 2 n x n = b2(1)

32

x 2 + 33 x3 + ... + 3n x n = b3 . .9

. m2( 2)

x 2 + m3 x3 + ... + mn x n = bm

(1)

(1)

(1)

(1)

a 22 0, m 1 (2). x2 n 2 (2). a11x1 + a12x2 + a13x3 + ... + a1nxn = b1 22 (3) x 2 + 23 x3 + ... + 2 n x n = b2 a33 x + . . . + 3 . . .( 2) ( 2) (1) (1) (1) (1) ( 2) ( 2)

a3n x = b3 n

a m 3 x + . . . + a mn x = bm 3 n n 1- a11x1 + a12x2 + a13x3 + ... + a1nxn = b1 22

( 2)

( 2)

x 2 + 23 x3 + ... + 2 n x n = b2( 2) ( 2) ( 2)

(1)

(1)

(1)

(1)

(4)

a33 x + . . . + a3n x = b3 3 n . . . ( n 1) ( a nn x n = bn n 1) a n +1,n( n 1) ( x n = bn n11) +

. . . a mn( n 1) ( x n = bmn 1)

k k- . (1) (4) a11, 22 , a33 , ... , a n 1 (1) ( 2) (1) ( 2) ( n2)

,n-1 0. , , a kk( n 2)

( k 1)

= 0, a kk

( k 1)

a11, 22 , a33 , ... , a n 1 ,n-1, , k- k-1 j- (j = ( n 1) ( k 1) ( k 1) a a k+1,k+2,...,m) jk . a kk = k +1,k = ...10

= a mk =0 (1) m n-1 .( ( bnn11) bnn 1) + ( n 1) ( n 1) a (1) m n a nn = n +1,n = . . . = ( bmn 1) (n a mn1) (5)

( k 1)

m>n (5) , (1) . m4 . 1. x+ y+z=2 2x y + 3z = 1 3x + 2y 3z =3 x, y, z. 1 1 1 1 1 1 1 1 2 1 3 2 1 3 2 1 D = 3 2 3 = 3 2 3 3 2 = 3 + 9 + 4 + 3 - 6 + 6 =19 Dx , Dy, Dz. 2 1 1 2 1 1 2 1 1 1 3 1 1 3 1 1 Dx = 3 2 3 = 3 2 3 3 2 = 6 + 9 + 2 + 3 12 + 3 =11 1 2 1 1 2 1 1 2 2 1 3 2 1 3 2 1 Dy = 3 3 3 = 3 3 3 3 3 = -3 +18 + 6 3 9 + 12 = 21 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 Dz = 3 2 3 = 3 2 3 3 2 = -3 + 3 + 8 + 6 2 6 = 6

D, Dx , Dy, Dz Dy Dx Dz 11 21 6 X = D = 19 Y = D = 19 Z = D = 19 2. , .14

x + y - 2z = 3 2x y z = 2 (I) (II) (III) x 5y + 4x = 6 D 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 D = 1 5 4 = 1 5 4 1 5 = -4 1 + 20 2 5 8 = 0

D , . : I, II, III, -3, 2, 2II 3I = III. 22 33 6 , , . 3. 2x 3y + 4z = 2 x + 5y z = -3 3x + 2y + 3z = -1 (I) (II) (III)

2 3 4 2 3 4 2 3 1 5 1 1 5 1 1 5 3 = 3 2 3 3 2 = 30 + 9 + 8 60 + 4 + 9 = 0 D= 3 2 D . I, II, III, I + II = III. , 2 3 = -1 . : 2x 3y = 2 - 4z x + 5y = -3 + z 1 17 z 6z 8 X = 13 Y = 13 1 17 z 6 z 8 (x,y,z)=( 13 , 13 ,z) z R x 2y z = 2 -2x + 4y + 4z =-415

4.

x 2 -yz=1 1 2 2 1 2 2 1 2 2 4 4 2 4 4 2 4 1 1 1 1 1 1 1 1 2 D= 2 = 2 =0

D , . -2 1 , 2 1 I, II, III, II = -2 I III = 2 I. 1 -4 = -22 1 = 2 2 . x = 2 + 2y + 2z. (x,y,z)=(2 + 2y + 2z,y,z) y,z R 5. .

2x + 6y + (a+6)z = 0 -x + 7y + 5 z = 0 ax + 5y + 13z = 0 2 6 a+6 1 7 5 D = a 5 13 = 0 -7a2 17a + 180 = 0 17 17 2 + 4 180 7 14 a1/2 =16

17 73 56 a1 = 14 = 14 = 4 17 + 73 90 45 a2 = 14 = 14 = - 7

45 = 7

a = 4 2x + 6y + 10z = 0 / (-2) -x + 7y + 5z = 0 4x + 5y + 13z = 0

45 7 x + 5y + 13z = 0

45 2x + 6y + z ( 7 +6)z = 0 -x + 7y + 5z = 0

2x + 6y + 10z = 0 / :2 -x + 7y + 5z = 0 -7y 7z = 0 2x + 6y + 10z = 0 10y + 10 = 0 /:10 -7y 7z = 0 / :(-7) 2x + 6y + 10z = 0 y + z = 0 / (-1) y+ z=0 2x + 6y + 10z = 0 y+ z=0 0=0 y = t, t R t+z=0 z = -t 2x + 6t 10t =0 2x = -6t + 10t17

45 7 x + 5y + 13z = 0 / 7 14x + 42y - 3z = 0 -x + 7y + 5z = 0 -45x + 35y + 91z = 0 -x + 7y + 5z = 0 / 14 /(-45) 14x + 42y - 3z = 0 -45x + 35y + 91z = 0 -x + 7y + 5z = 0 140y + 67z = 0 / 2 -280y - 134z = 0 -x + 7y + 5z = 0 140y + 67z = 0 0=0 y = t, t R 67z=-140y

3 2x + 6y - 7 z = 0 / 7 -x + 7y + 5z = 0

2x = 4t / :2 x = 2t (x,y,z) = (2t,t,-t)

140 z= 67 t -x = -7y -5z x = 7y + 5z

140 x = 7t + 5( 67 t) 700 x =7t - 67 t 7 67t 700t 231 67 x= = 67 t 231 140 (x,y,z)=( 67 t,t, 67 t )

6. . x + y + az + t = 0 x y z t =0 ax+ y + 5z + 3t = 0 x+ 5y+11z+ 8t = 0 1 1 a 1 1 1 1 1 a 1 5 3 D = 1 5 11 8 = 1 2 a +1 2 2 a +1 2 1 0 0 0 a +1 5 + a a + 3 a a +1 5 + a a + 3 12 9 1 6 12 9 = -1 6 =

= - (18(5+ a) +24(a+1) + 6(a+1)(a+3) 12(5+a) 24(a+3) 9(a+1)2) = = -(90 + 18a + 24a +24+6a2+24a+18-60-12a-24a-72-9a2 -9-18a) = -(-3a2 +12a -9)= 3a2-12a+9 D=0 3a2-12a+9=0 /:3 a2-4a+3=0 (a-1)(a-3)=0 a=1 a=3 3c x+c- 2 +0=0 c x- 2 =0 c x= 2

a =3 x + y + 3z + t = 0 /(-1) /(-3) /(-1) x y zt =0 3x+ y+5z+3t = 0 x +5 y + 11z +8 t = 0

18

x + y + 3z + t = 0 2 y 4 z 2 t = 0 /(-1) / 2 2 y 4 z + 0 = 0 4 y +8 z + 7t = 0 x + y + 3z + t = 0 2 y 4 z 2 t = 0 2t=0 3t = 0 x + y + 3z + t = 0 2 y 4 z 2 t = 0 t=0 y = c , c R -2c 4z = 0 4z = 2 c c z= 2 a =1 x + y + z + t = 0 / (-1) / (-1) / (-1) x y zt =0 x + y + 5z + 3t = 0 x +5 y + 11z +8 t = 0 x+ y+ z + t=0 2 y 2 z 2 t =0 / 2 4z + 2t = 0 4y + 10z + 7t=0 x+ y+ z + t=0 2 y 2 z 2 t = 0 4z + 2t = 0 / 3 6z + 3t = 0 / (-2) x+ y+ z + t=0 2 y 2 z 2 t =0 12z + 6t = 0 -12z - 6t = 0 x+ y+ z + t=0 2 y 2 z 2 t =0 12z + 6t = 0 0=0 t = d, d R 12z + 6d = 0 12z = - 6d / :12

c c ( x,y,z.t)=( 2 ,c,- 2 ,0)

c R

19

1 z=-2d d d x- 2 - 2 +d=0 xd+d=0 x=0 d -2y 2 (- 2 ) 2d = 0 -2y + d 2d = 0 -2y = d 1 y=-2d d d (x,y,z,t) = (0, 2 , 2 ,d) d R

7. a. ax + y + z = 1 x + ay + 2z = 2 2x + y + z = 0 a 1 1 1 a 2 D = 2 1 1 = a2 + 4 +1 2a 2a 1 = a2 4a + 4 = (a-2)2 Dx D y Dz D 0 (x,y,z)=( D , D , D ) (a-2)2 0 a 2 1 1 1 2 a 2 Dx= 0 1 1 = a 2 a 1 1 1 2 2 Dy= 2 0 1 = 2a + 4 4 1 = 2a 1 a 1 1 1 a 2 Dz= 2 1 0 = 4 + 1 2a 2a = 5 4a20

a2 2a 1 5 4a Dy Dx Dz 1 2 2 2 X = D = (a 2) = a 2 ; Y = D = (a 2) ; Z = D = (a 2) 2a 1 5 4a 1 2 2 (x,y,z)=( a 2 , (a 2) , (a 2) )

D = 0 (a-2)2 = 0 , a = 2 2x + y + z = 1 /(-1) x + 2y + 2z = 2 2x + y + z = 0 2x + y + z = 1 x + 2y + 2z = 2 0 = -1 a = 2 .

8. . ax + y - z = x + ay - z =1 x - y - z = a 1 1 1 a 1 D = 1 1 a = -a3 1 + 1 + a a + a = -a3 + a = a(1-a2)=a(1-a)(1+a) D0 . 0 ; 1 ; -1

21

a 1 1 1 a 1 Dx= a 1 a = -a3 +1 a + a2 a + a = a2 + 1 a(a2+1)=( a2+1)(1-a) a a 1 1 1 1 Dy= 1 a a = -a2 a a + 1 + a2 + a2 = a2 2a +1= (1-a)2 a 1 a 1 a 1 Dz= 1 1 a = a3 +1 a a2 + a a = (a3 a ) + (1 a2) = -a(-a2 + 1) + (1 a2)= = (1 a2)(1-a)= (1-a)(1+a)(1-a)=(1-a)2(1+a)2 2 (1 a) 2 1 a Dy D x (a + 1)(1 a) a + 1 X = D = a (1 a )(1 + a ) = a (1 + a ) ; Y = D = a (1 a)(1 + a ) = a (1 + a ) ; 2 D z (1 + a )(1 a) 1 a Z = D = a (1 a)(1 + a ) = a

a2 +1 1 a 1 a (x,y,z)=( a (1 + a ) , a (1 + a ) , a ) 0 1

=0 y - z =0 x - z =1 x- y =0 x - z =1 / (-1) x- y =0 y - z =0 x - z =1 -y + z =-1 y - z =0 =-1 -x + y - z = -1 x - y - z =1 x - y + z = -1 -x + y - z = -1 -2z = 0 0 = -2 =-1 =1 x + y - z = 1 / (-1)/(-1) x+ y- z=1 x- y - z=1 x+ y- z=1 0=0 -2y = 0 y=0 x - z =1 z = t, t R x=1+t (x,y,z)=( 1 + t,0,t)

x

- z =1 -y + z =-1 0=-1 =0

22

. . . , , , . , . , Carl Friedrich Gauss, : ,, , . , . Disquisitiones arithmeticae , , .

, ,, . . ,, , . , 3

23

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........................................................................5 .............5 .......................6 ........................................................................... .....................................13 ........................................................................... ......................................18 ............................................................................................................. 20

25