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)
I did this for you when you were Saphina AlMatary
http://www.jiskha.com/display.cgi?id=1477322557
I gave you a good hint for the second one
Did you even look at it?
In (a),
y=4z+25
so, use that in (1) and (2). Then you just have to solve two equations.
Similarly in (b), #3 gives
y = 3x+2z+7
Use that in (1) and (2)
To solve simultaneous equations, we can use the method of substitution or elimination. I will explain the method of elimination for both sets of equations.
(a)
To solve this set of equations using elimination, we can eliminate one variable at a time.
Step 1: Start by multiplying one or both equations by a suitable constant so that when added or subtracted, one variable will cancel out.
In this case, we can eliminate the variable x by multiplying equation (2) by -1 and equation (3) by 2.
Equation (2) multiplied by -1: -x + y + 2z = -13 (4)
Equation (3) multiplied by 2: 2y - 8z = 50 (5)
Step 2: Now, we can eliminate the variable y by multiplying equation (4) by 2 and equation (5) by 1.
Equation (4) multiplied by 2: -2x + 2y + 4z = -26 (6)
Equation (5) multiplied by 1: 2y - 8z = 50 (7)
Step 3: Add or subtract the equations to eliminate the variable y.
Equation (6) + Equation (7): -2x + 2y + 4z + 2y - 8z = -26 + 50
4z = 24
z = 6
Step 4: Substitute the value of z back into one of the original equations to solve for y.
Using equation (3): y - 4z = 25
y - 4(6) = 25
y - 24 = 25
y = 49
Step 5: Substitute the values of y and z back into one of the original equations to solve for x.
Using equation (1): x + y + z = -3
x + 49 + 6 = -3
x + 55 = -3
x = -58
So, the solution to the simultaneous equations in (a) is x = -58, y = 49, z = 6.
(b)
Following the same steps as above, we can solve this set of equations using elimination.
Step 1: Multiply equation (1) by 2, equation (2) by -1, and equation (3) by -3.
Equation (1) multiplied by 2: 2x + 4y + 6z = -28 (4)
Equation (2) multiplied by -1: -2x - 3y + z = -13 (5)
Equation (3) multiplied by -3: -9x + 3y - 6z = 21 (6)
Step 2: Add or subtract the equations to eliminate the variable x.
Equation (4) + Equation (5): 2x + 4y + 6z - 2x - 3y + z = -28 - 13
y + 7z = -41 (7)
Equation (7) + Equation (6): y + 7z - 9x + 3y - 6z = -41 + 21
-9x + 4y + z = -20 (8)
Step 3: Add or subtract the equations to eliminate the variable y.
Equation (8) - Equation (5): -9x + 4y + z - (-2x - 3y + z) = -20 - (-13)
-7x + 7y = -7
-x + y = -1 (9)
Step 4: Substitute the value of y back into the equation (9) to solve for x.
Using equation (9): -x + y = -1
-x + (-x + 1) = -1
-2x + 1 = -1
-2x = -2
x = 1
Step 5: Substitute the value of x back into equation (7) to solve for z.
Using equation (7): y + 7z = -41
(-1) + 7z = -41
7z = -40
z = -40/7 or -5.71
Step 6: Substitute the values of x and z back into equation (1) to solve for y.
Using equation (1): x + 2y + 3z = -14
1 + 2y + 3(-40/7) = -14
1 + 2y - 120/7 = -14
2y - 120/7 = -15
2y = -15 + 120/7
2y = -105/7 + 120/7
2y = 15/7
y = 15/7
So, the solution to the simultaneous equations in (b) is x = 1, y = 15/7, z = -40/7 or approximately x = 1, y = 2.14, z = -5.71.