Solve the following system of equations
1. X+y-z= 6
2x-y +z=-9
X-2y+3z=1
2. x+z=-3
y+z=3
x+y=8
1. 1st plus 2nd ... 3x = 3
twice the 1st plus 3rd ... 3x + z = 13
2. 1st minus 2nd plus 3rd ... 2x = 2
x+y-z= 6
2x-y +z=-9
X-2y+3z=1
using substitution, you can do:
from #3,
x = 2y-3z+1
Using that in #1 and #2, you get
(2y-3z+1)+y-z = 6
2(2y-3z+1)-y+z = -9
3y-4z = 5
3y-5z = -11
Now, using 3y=4z+5,
(4z+5)-5z = -11
-z = -16
z = 16
3y = 4*16+5 = 69, so y = 23
x = 2*23-3*16+1 = -1
and similarly for the 2nd problem
To solve the first system of equations:
Step 1: We will use the method of substitution to solve this system.
Start by solving equation (1) for x:
x = 6 - y + z
Step 2: Substitute the value of x into the other two equations to create a new system of equations.
Equation (2):
2(6 - y + z) - y + z = -9
12 - 2y + 2z - y + z = -9
12 - 3y + 3z = -9
Equation (3):
(6 - y + z) - 2y + 3z = 1
6 - y + z - 2y + 3z = 1
6 - 3y + 4z = 1
Now we have the following system of equations:
12 - 3y + 3z = -9 (4)
6 - 3y + 4z = 1 (5)
Step 3: Solve equations (4) and (5) simultaneously to obtain the values of y and z.
To do this, we'll subtract equation (5) from equation (4):
(12 - 3y + 3z) - (6 - 3y + 4z) = -9 - 1
12 - 3y + 3z - 6 + 3y - 4z = -10
-2z = -10
z = -10 / -2
z = 5
Substitute the value of z into equation (4) to solve for y:
12 - 3y + 3(5) = -9
12 - 3y + 15 = -9
-3y = -9 - 12 - 15
-3y = -36
y = -36 / -3
y = 12
Step 4: Substitute the values of y and z back into equation (1) to solve for x:
x + 12 - 5 = 6
x + 7 = 6
x = 6 - 7
x = -1
Therefore, the solution to the first system of equations is x = -1, y = 12, z = 5.
To solve the second system of equations:
Step 1: We will use the method of substitution again to solve this system.
Step 2: Solve equation (1) for x:
x = -3 - z
Step 3: Substitute the value of x into the other two equations to create a new system of equations.
Equation (2):
(-3 - z) + z = 3
-3 = 3
This equation is inconsistent and has no solution.
Therefore, the second system of equations has no solution.
1.
x + y - z = 6
+
2 x - y + z = - 9
______________
3 x = - 3
Divide both sides by 3
x = - 1
2 x - y + z = - 9 Multiply both sides by - 3
- 6 x + 3 y - 3 z = 27
- 6 x + 3 y - 3 z = 27
+
x - 2 y + 3 z = 1
_________________
- 5 x + y = 28
- 5 ∙ ( - 1 ) + y = 28
5 + y = 28
Subtract 5 to both sides
5 + y - 5 = 28 - 5
y = 23
x + y - z = 6
- 1 + 23 - z = 6
22 - z = 6
Subtract 22 to both sides
22 - z - 22 = 6 - 22
- z = - 16
Multiply both sides by - 1
z = 16
The solutions are:
x = - 1 , y = 23 , z = 16
2.
x + z = - 3
Subtract z to both sides
x + z - z = - 3 - z
x = - 3 - z
x + y = 8
Subtract y to both sides
x + y - y = 8 - y
x = 8 - y
x = x
- 3 - z = 8 - y
Add 3 to b oth sides
- 3 - z + 3 = 8 - y + 3
- z = 11 - y
Multiply both sides by - 1
z = - 11 + y
z = y - 11
Now:
y + z = 3
y + y - 11 = 3
2 y - 11 = 3
Add 11 to both sides
2 y - 11 + 11 = 3 + 11
2 y = 14
Divide both sides by 2
y = 14 / 2
y = 7
x = 8 - y
x = 8 - 7
x = 1
z = y - 11
z = 7 - 11
z = - 4
The solutions are:
x = 1 , y = 7 , z = - 4