Problem 1 :
Solve the equations
x+2y+3z = 14, 3x+y+2z = 11, 2x+3y+z = 11
Solution :
x+2y+3z = 14 --------(1)
3x+y+2z = 11 --------(2)
2x+3y+z = 11 --------(3)
(1)-3(3)
x+2y+3z - 3(2x+3y+z) = 14 - 3(11)
x+2y+3z - 6x-9y-3z = 14 - 33
-5x-7y = -19 ----(4)
(2)-2(3)
3x+y+2z-2(2x+3y+z) = 11-2(11)
3x+y+2z-4x-6y-2z = 11-22
-x-5y = -11 ----(5)
(4) - 5(5)
-5x-7y-5(-x-5y) = -19-5(-11)
-5x-7y+5x+25y = -19+55
-7y+25y = 36
18y = 36
y = 2
By applying the value of y in (5), we get
-x-5(2) = -11
-x-10 = -11
-x = -1
x = 1
By applying the value of x and y in (1), we get
1+2(2)+3z = 14
5+3z = 14
3z = 9
z = 3
So, the solution is x = 1, y = 2 and z = 3.
Problem 2 :
Solve the equations
3x-3y+4z = 14, -9x-6y+2z = 1, 6x+3y+z = 5
Solution :
3x-3y+4z = 14 ----(1)
-9x-6y+2z = 1 ----(2)
6x+3y+z = 5 ----(3)
2(1) - (2)
2(3x-3y+4z)-(-9x-6y+2z) = 2(14) - 1
6x-6y+8z+9x+6y-2z = 28-1
15x+6z = 27 -----(4)
(2) + 2(3)
-9x-6y+2z+2(6x+3y+z) = 1 + 2⋅5
-9x-6y+2z+12x+6y+2z = 1 + 10
3x+4z = 11 -----(5)
2(4) - 3(5)
2(15x+6z)-3(3x+4z) = 2⋅27 - 3⋅11
30x+12z-9x-12z = 54-33
21x = 21
x = 1
By applying the value of x in (5), we get
3(1)+4z = 11
3+4z = 11
4z = 8
z = 2
By applying the value of x and z in (1), we get
3(1)-3y+4(2) = 14
3-3y+8 = 14
-3y+11 = 14
-3y = 3
y = -1
Therefore solution is x = 1, y = -1 and z = 2
Problem 3 :
Solve the equations
3x-2y+z = 0, 4x+6y-3z = 13, x-2y+2z = -4
Solution :
3x-2y+z = 0 -----(1)
4x+6y-3z = 13 -----(2)
x-2y+2z = -4 -----(3)
3(1) + (2)
3(3x-2y+z)+(4x+6y-3z) = 0+13
9x-6y+3z+4x+6y-3z = 13
9x+4x = 13
13x = 13
x = 1
(2) + 3(3)
(4x+6y-3z)+3(x-2y+2z) = 13+3⋅(-4)
4x+6y-3z+3x-6y+6z = 13-12
7x+3z = 1 ----(4)
By applying the value of x in (4), we get
7(1)+3z = 1
3z = -6
z = -2
By applying the value of x and z in (1), we get
3(1)-2y-2 = 0
3-2-2y = 0
-2y = -1
y = 1/2
Therefore solution is x = 1, y = 1/2 and z = -2
Problem 4 :
Solve the equations
2x-2y+4z = -12, 3x+2y+2z = 19, -x+y-z = 3
Solution :
2x-2y+4z = -12 -----(1)
3x+2y+2z = 19 -----(2)
-x+y-z = 3 -----(3)
(1)+(2)
(2x-2y+4z)+(3x+2y+2z) = -12+19
5x+6z = 7 ------(4)
(2)-2(3)
(3x+2y+2z)-2(-x+y-z) = 19-2⋅ 3
3x+2y+2z+2x-2y+2z = 19-6
5x+4z = 13 ------(5)
(4)-(5)
5x+6z - (5x+4z) = 7-13
2z = -6
z = -3
By applying the value of z in (4), we get
5x+4(-3) = 13
5x = 13+12
5x = 25
x = 5
By applying the value of x and z in (3), we get
-5+y-(-3) = 3
-5+y+3 = 3
-2+y = 3
y = 5
Therefore solution is x = 5, y = 5 and z = -3.
Problem 5 :
Solve the equations
2x+3y-z = 5, 4x+y+3z = 5, 3x+2y+2z = 5
Solution :
2x+3y-z = 5 ----(1)
4x+y+3z = 5 ----(2)
3x+2y+2z = 5----(3)
(1)-3(2)
(2x+3y-z)-3(4x+y+3z) = 5-3⋅5
2x-12x+3y-3y-z-9z = 5 - 15
-10x-10z = -10
x+z = 1----(4)
2(2)-(3)
2(4x+y+3z)-(3x+2y+2z) = 2⋅5-5
8x+2y+6z -3x-2y-2z = 5
5x+4z = 5 ---(5)
4(4)-(5)
4x+4z-5x-4z = 4-5
-x = -1
x = 1
By applying the value of x in (1), we get
5(1)+4z = 5
4z = 0
z = 0
By applying the value of z and x in (1), we get
2(1)+3y-0 = 5
3y = 3
y = 1
Therefore solution is x = 1, y = 1 and z = 0.
Problem 6 :
Solve the equations
x+2y+3z = 10, x-2y+4z = 3, x+y–3z = 2
Solution :
x+2y+3z = 10 ----(1)
x-2y+4z = 3 ----(2)
x+y–3z = 2 ----(3)
(1) + (2)
(x+2y+3z)+(x-2y+4z) = 10+3
2x+7z = 13 ----(4)
(2)+2(3)
(x-2y+4z)+2(x+y–3z) = 3+4
x+2x-2y+2y+4z-6z = 7
3x-2z = 7 ----(5)
2(4)+7(5)
2(2x+7z)+7(3x-2z) = 26+7⋅7
4x+14z+21x-14z = 26+49
25x = 75
x = 3
By applying the value of x in (3), we get
3(3)-2z = 7
9-2z = 7
-2z = 7-9
-2z = -2
z = 1
By applying the value of z and x in (1), we get
3+2y+3(1) = 10
6+2y = 10
2y = 4
y = 2
Therefore solution is x = 3, y = 2 and z = 1.
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