1) Pour chaque système, discuter de l'existence de solutions. 2) Résoudre les systèmes admettant une unique solution. a) 6x-5y=38 3x-y=13 b) 2x-3y=-2 2x-5y c) 4

a) 6x-5y=38
3x-y=13

3x = 13 +y
x = 13/3 +1/3y

6x -5y =38
6 (13/3 +1/3y) -5y =38
26 +2y -5y =38
-3y = 38-26
y = 12/-3
y = -4

x = 13/3 +1/3 *-4
x =13/3 -4/3
x = 9/3
x = 3

b) 2x-3y=-2
2x-5y

il manque pas une information pour la 2eme equation 2x-5y = ?

c) 4x+9y=35
6x-5y=-77

6x = -77 +5y
x = -77/6 +5/6y

4x+9y =35
4(-77/6 +5/6y) +9y =35
-144/3 +10/3y +9y =35
10/3 y +27/3 y = 35 +144/3
37/3 y = 105/3 +144/3
37/3y = 249/3
y = 83 x3/37
y = 249/37

x = -77/6 +5/6 *249/37
x = -77/6 +1245/222
x =-2849/222 + 1245/222
x = -1604 /222
x =-802/111

d) 8y-6x=8
-4x+3y=-1

-4x = 1-3y
x = -1/4 +3/4 y

8y -6(-1/4+3/4y) =8
8y +3/2 -9/2y = 8
8y -9/2y = 8 -3/2
16/2y -9/2y = 16/2-3/2
7/2y = 13/2
y =13/2 x2/7
y =13/7

x =1/4 +3/4 *13/7
x =1/4 +39/28
x =7/28 +39/28
x =46/28
x =23/14

a)
6x-5y=38
3x-y=13 ⇒ ×(-5)

6x - 5y -15x +5y = 38-65
-9x = -27
x = 27/9 = 3

3x-y=13
-y = -9 +13
y = -4

S= {3 ; -4}

b)
2x-3y=-2 ⇒ ×5
2x-5y= -7 ⇒ ×(-3)

10x - 15y -6x +15y = -10 + 21
4x = 11
x = 11/4 = 2,75

2x-3y=-2
2*(2,75) -3y = -2
-3y = -7,5
y = 7,5/3 = 2,5

S= {2,75 ; 2,5}

c)
4x+9y=35 ⇒ ×5
6x-5y=-77 ⇒ ×9

20x +45y +54x -45y= 175 -693
74x = -518
x = -518/74 = -7

4x+9y=35
4*(-7) +9y = 35
9y = 63
y = 7

S= {-7 ; 7}

d)
8y-6x=8 ⇒ ×(-3)
-4x+3y=-1 ⇒ ×8

-24y +18x -32x +24y= -24-8
-56x = -32
x = 32/56 = 4/7

8y-6x=8
8y -6*(4/7) = 8
8y = 80/7
y = (80/7)/8 = 10/7

S= {4/7 ; 10/7}