ECG Mathématiques Approfondies: TD 09 Systèmes Linéaires (2025-2026)

1.
x+y+z= 1
y+z= 2
z= 3
2.
2x+ 4y−2z= 0
2y+ 3z= 2
2z= 4
3.
x+y+z= 1
3x+y+ 7z= 17
x+ 2y+ 2z= 3
1.
4x+ 11z= 2
2x+y+ 6z= 2
−x+y−2z= 0
2.
−y+ 3z= 1
x−y+ 4z= 2
−x+y−3z= 3
3.
2x−5y+ 2z=−14
−x+ 3y+z= 9
−y+z=−3
1) x−y+3z= 3
−x+y−z= 1 2) x+2y−z=−1
−x−y+2z= 2 3) 2x−4y= 2
−3x+6y= 6
4) 
x+2y−z= 1
2x−y+z=−1
x+7y−4z= 4
5) 
2x−3y−2z= 1
2x+y+2z= 9
6x−4y−z= 8
6) 
3x−2y+z= 7
x+2y−5z= 6
x−y+z= 2
k∈
kx +y= 1
x+ky = 1
⋄
3x−y−12z= 6
3y−16z= 8
−3x+5y−9z= 4 
2x+3y+2z=−4
8x+8y+10z=−11
4x+4y+6z=−7
6x+6y= 10
9x+9y−z= 17
6x+6y= 10 
2x−4y−7z=−11
2x−4y−7z=−14
−2x+4y+5z=−1
2x−6y−11z= 12
−2x+8y+15z=−16
−2y−3z= 1

λ∈
λx +y+z= 1
x+λy +z=λ
x+y+λz =λ2⋄
m∈
(S1)
mx +y+z=A
x+my +z=B
x+y+mz =C
(S2)(m+ 1)x+my = 2m
mx + (m+ 1)y= 1
λ
(Sλ)
(2 −λ)x+y−7z= 0
2x+ (3 −λ)y−8z= 0
2x+ 2y−(7 + λ)z= 0
P(a, b, c)
P(x) = ax2+bx +c
P(a, b, c, d)
P(x) = ax3+bx2+cx +d⋄
P2
P(0) = 0 P(1) = 1 P(2) = 2.
P2
P′(0) = 3 P′(1) = 1 P(2) = 2.
P3
P(0) = 0 P′(1) = 1 P′(2) = 2 P′′(0) = 2.

1
x+ey+ ln(z) = 1
3
x−ey+ ln(z) = 0
−2
x+ 2ey+ 3 ln 1 z= 3
⋄
m∈
1) 
x+2y−z= 3
−x+(m−2)y+2z=m−2
−2x+(2m−4)y+2z= 2m−2
2) 
x+y−z= 3
mx +y−(m+ 1)z= 3m
mx +(2m−1)y+2z= 4m+ 3
x−2y−2z−4t= 2
x−2y−1z−2t= 4
−x+y+2z+2t= 0
1.2x−3y+7z= 5
x+4y−3z= 7 2.x+2y−z=−1
−x−y+2z= 2
3.
u+w= 1
v+w= 0
u+v= 1
2u+3v= 0
4.
x+y+z= 3
y+z+t= 3
x+z+t= 3
x+y+t= 3
3x+2y= 3
5x−y= 2 −3x+2y= 3
6x−4y= 2 (√2−1)x+y= 3
x+(√2 + 1)y= 2
√3x+3y= 3
3x+3√3y= 2 √5x+2y= 3
3x+4y= 2 3x+2y= 3
x−y= 2

2x−3y−8z−t=−11
−x+2y+5z+t= 6
−2x+3y+9z−t= 13
3x−2y+z= 7
x+2y−5z= 6
x−y+z= 2
1.2x−y+ 3z= 9
3x−5y+z=−42.4x−2y= 3
4x−2y+z=−13.
2x−y+ 3z= 9
3x−5y+z=−4
4x−7y+z= 5
4.
2x+ 3y−z= 2
5x−2y+ 3z= 5
x−3y+z=−1
5.
2x+ 3y−z= 5
5x−2y+ 3z=−2
x+y−z= 1
x−3y+z=−1
3x+ 3z=−2
6.
−x+ 5y−z= 3
3x−y+ 3z=−1
2x+ 2y−z= 2
4x+ 6y+z= 4
7.
2x−y+z= 3
x+y−2z=−3
4x+ 2y−4z=−4
2v−w= 1
−2u+ 4v+ 3w=−1
u+v−3w=−6
x+y+z+t= 2
2x+y+z+t= 1
x+ 2y+ 2z= 2
u+w= 1
v+w= 0
u+v= 1
2u+ 3v= 0
x+ 2y−z=a1
−2x−3y+ 3z=a2
x+y−2z=a3
(x, y, z)
(a1, a2, a3) = (0,0,0) (a1, a2, a3) = (1,−2,1) (a1, a2, a3) = (1,2,1) ⋄
4x+12y+3z=−7
7x+21y+6z=−12
12x+36y+10z=−21