Solve the simultaneous questions:
(a) x+y+z=-3 (1)
x-y-2z=13(2)
y-4z=25 (3)
(b) x+2y+3z=-14 (1)
2x+3y-z=13 (2)
3x-y+2z=-7 (3)
To solve these simultaneous equations, we can use the method of elimination or substitution. Let's solve both sets of equations step by step:
(a) x+y+z=-3 (1)
x-y-2z=13 (2)
y-4z=25 (3)
Method 1: Substitution
Step 1: Solve equation (3) for y in terms of z.
From equation (3):
y = 4z + 25
Step 2: Substitute y from equation (3) into equations (1) and (2).
Substituting y into equation (1):
x + (4z + 25) + z = -3
x + 5z + 25 = -3
x + 5z = -3 - 25
x + 5z = -28 (4)
Substituting y into equation (2):
x - (4z + 25) - 2z = 13
x - 4z - 25 - 2z = 13
x - 6z - 25 = 13
x - 6z = 13 + 25
x - 6z = 38 (5)
Now we have two equations with two variables, (4) and (5).
Step 3: Solve equations (4) and (5) simultaneously for x and z.
Multiplying equation (4) by 6:
6x + 30z = -168 (6)
Multiplying equation (5) by -1:
-x + 6z = -38 (7)
Adding equations (6) and (7) eliminates x:
6x + 30z + (-x + 6z) = -168 + (-38)
5x + 36z = -206 (8)
Dividing equation (8) by 5:
x + 7.2z = -41.2 (9)
We can now solve equations (9) and (7) simultaneously.
Multiplying equation (9) by 6:
6x + 43.2z = -247.2 (10)
Multiplying equation (7) by -6:
6x - 36z = 228 (11)
Adding equations (10) and (11) eliminates x:
6x + 43.2z + (6x - 36z) = -247.2 + 228
12x + 7.2z = -19.2
Dividing equation (12) by 12:
x + 0.6z = -1.6 (13)
Now we have the value of x in terms of z.
Step 4: Solve equations (13) and (7) simultaneously for x and z.
Multiplying equation (13) by -1:
-x - 0.6z = 1.6 (14)
Multiplying equation (7) by 1:
-x + 6z = -38 (15)
Adding equations (14) and (15) eliminates x:
-x + 6z + (-x - 0.6z) = -38 + 1.6
5.4z = -36.4
Dividing equation (16) by 5.4:
z = -6.741
Now we have the value of z.
Step 5: Substitute z = -6.741 into equation (4) to find x:
x + 5(-6.741) = -28
x - 33.705 = -28
x = -28 + 33.705
x = 5.705
Step 6: Substitute x = 5.705 and z = -6.741 into equation (3) to find y:
y - 4(-6.741) = 25
y + 26.964 = 25
y = 25 - 26.964
y = -1.964
Therefore, the solution to the simultaneous equations (a) is:
x = 5.705, y = -1.964, z = -6.741
Now let's solve the second set of equations, (b):
(b) x+2y+3z=-14 (1)
2x+3y-z=13 (2)
3x-y+2z=-7 (3)
Method 2: Elimination
Step 1: Multiply equation (1) by 3 and equation (3) by 2:
3x + 6y + 9z = -42 (4)
6x - 2y + 4z = -14 (5)
Step 2: Add equations (4) and (5) to eliminate x:
(3x + 6y + 9z) + (6x - 2y + 4z) = -42 + (-14)
9x + 4y + 13z = -56 (6)
Now we have two equations with two variables, (5) and (6).
Step 3: Multiply equation (2) by 3 and equation (3) by 2:
6x + 9y - 3z = 39 (7)
6x - 2y + 4z = -14 (8)
Step 4: Subtract equation (7) from equation (8) to eliminate x:
(6x - 2y + 4z) - (6x + 9y - 3z) = -14 - 39
-11y + 7z = -53 (9)
Now we have two equations with two variables, (6) and (9).
Step 5: Multiply equation (6) by 11 and equation (9) by 9:
99x + 44y + 143z = -616 (10)
-99y + 63z = -477 (11)
Step 6: Add equations (10) and (11) to eliminate y:
(99x + 44y + 143z) + (-99y + 63z) = -616 + (-477)
99x + 107z = -1093 (12)
Now we have one equation with one variable, (12).
Step 7: Solve equation (12) for x in terms of z:
99x = -1093 - 107z
x = (-1093 - 107z)/99 (13)
Step 8: Substitute x from equation (13) into equation (6):
9((-1093 - 107z)/99) + 4y + 13z = -56
Simplifying and rearranging the equation:
-121z - 396 = 0
Solving for z:
z = (-396)/(-121)
z = 3.273
Now we have the value of z.
Step 9: Substitute z = 3.273 into equation (9):
-11y + 7(3.273) = -53
Simplifying and rearranging the equation:
-11y + 22.911 = -53
-11y = -53 - 22.911
-11y = -75.911
y = (-75.911)/(-11)
y = 6.901
Now we have the value of y.
Step 10: Substitute y = 6.901 and z = 3.273 into equation (13):
x = (-1093 - 107(3.273))/99
x = (-1093 - 351.611)/99
x = (-1444.611)/99
x = -14.593
Therefore, the solution to the simultaneous equations (b) is:
x = -14.593, y = 6.901, z = 3.273
Note: The solutions are approximate values.