To solve the system of equations:
Equation 1: x - 3y + 2z = 11
Equation 2: -x + 4y + 3z = 5
Equation 3: 2x - 2y - 4z = 2
First, we'll use the method of elimination to eliminate variables one by one.
Multiplying Equation 1 by 2 gives us:
2x - 6y + 4z = 22 -> Equation 4
Adding Equation 3 to Equation 4 eliminates the x variable:
(2x - 2y - 4z) + (2x - 6y + 4z) = 2 + 22
4x - 8y = 24 -> Equation 5
Next, multiplying Equation 3 by 3 gives us:
6x - 6y - 12z = 6 -> Equation 6
Adding Equation 5 to Equation 6 eliminates the y variable:
(4x - 8y) + (6x - 6y - 12z) = 24 + 6
10x - 14z = 30 -> Equation 7
Multiplying Equation 2 by 2 gives us:
-2x + 8y + 6z = 10 -> Equation 8
Adding Equation 1 to Equation 8 eliminates the y variable:
(x - 3y + 2z) + (-2x + 8y + 6z) = 11 + 10
- x + 8z = 21 -> Equation 9
Simplifying Equation 7 gives us:
5x - 7z = 15 -> Equation 10
Now we have a system of two equations to solve simultaneously:
Equation 10: 5x - 7z = 15
Equation 9: -x + 8z = 21
Multiplying Equation 9 by 5 and Equation 10 by -1 gives us:
-5x + 40z = 105 -> Equation 11
-5x + 7z = -15 -> Equation 12
Adding Equation 11 to Equation 12 eliminates the x variable:
(-5x + 40z) + (-5x + 7z) = 105 - 15
47z = 90
Solving for z gives us:
z = 90/47
Substituting z back into Equation 10:
5x - 7(90/47) = 15
Simplifying:
5x - (630/47) = 15
5x = 15 + (630/47)
5x = 705/47
x = (705/47) / 5
x = 705/235
x = 3
Substituting the values of x and z into Equation 9:
-3 + 8(90/47) = 21
Simplifying:
3 - (720/47) = 21
- (720/47) = 21 - 3
- (720/47) = 18
720/47 = -18
Therefore, the solution to the system of equations is:
x = 3
y = -18
z = 90/47
what is the solution to the following systems of equations ? x-3y+2z=11 -x+4y+3z=5 2x-2y-4z=2
1 answer