Rezolvaţi prin metoda substituţiei următoarele sisteme: a) 2x + y = 8 - 2x, \qquad 3(x - y) - 2y = 29 b) 2x - 5y = 11, \qquad 3(x + y) - 2x = 11 d) 3(x + 4) + y = 3, \qquad x + 2(y + 5) = 2 e) 4x - 3y = 13, \qquad 2(x - y) + 5y = - 7 c) 2(x + 3) - 3y = 2, \qquad 5x - 3(2y - 1) = - 4 f) 5(x + 2) - 3 = y, \qquad x + 2(y + 1) = 5
Răspunsuri la întrebare
Răspuns:
a) x = 3; y = -4; b) x = 8; y = 1; c) x = 1; y = 2;
d) x = -2; y = -3; e) x = 1; y = -3; f) x = -1; y = 2
Explicație pas cu pas:
a) 2x + y = 8 - 2x => y = 8 - 4x
3(x - y) - 2y = 29 => 3x - 5y = 29
3x - 5(8 - 4x) = 29
3x - 40 + 20x = 29
23x = 69 => x = 3
y = 8 - 12 => y = -4
b) 2x - 5y = 11
3(x + y) - 2x = 11 => x + 3y = 11 => x = 11 - 3y
2(11 - 3y) - 5y = 11
22 - 6y - 5y = 11
11y = 11 => y = 1
x = 11 - 3 => x = 8
c) 2(x + 3) - 3y = 2 => 2x - 3y = -4
x = (3y - 4)/2
5x - 3(2y - 1) = - 4 => 5x - 6y = -7
5(3y - 4)/2 - 6y = -7
15y - 20 - 12y = -14
3y = 6 => y = 2
x = (6 - 4)/2 => x = 1
d) 3(x + 4) + y = 3 => y = - 3x - 9
x + 2(y + 5) = 2 => x + 2y = - 8
x + 2(-3x -9) = - 8
x - 6x - 18 = - 8
-5x = 10 => x = -2
y = -3(-2) - 9 => y = -3
e) 4x - 3y = 13 => x = (3y + 13)/4
2(x - y) + 5y = - 7
2x - 2y + 5y = -7 => 2x + 3y = -7
2(3y + 13)/4 + 3y = -7
2(3y + 13) + 12y = -28
6y + 26 + 12y = -28
18y = - 54 => y = -3
x = (-9 + 13)/4 => x = 1
f) 5(x + 2) - 3 = y => 5x + 7 = y
x + 2(y + 1) = 5 => x + 2y = 3
x + 2(5x + 7) = 3
x + 10x + 14 = 3
11x = - 11 => x = -1
y = -5 + 7 => y = 2