Solve the following system of equations:
x + 2y - z = 2
-3x + y + z = -9
x + y - 2z = 5
I came up with
(-2,-11/3, -2)
Is this correct?
No,
had you simply substituted your answer into the 1st equation, you would have seen that it was not correct
Add the 1st and 2nd ---> -2x+3y = -7 , (#4)
double the 2nd , then add to the 3rd ---> -5x +3y =-13 , (#5)
#4 - #5 :
3x = 6
x = 2
sub back into #4
-4 + 3y = -7
3y = -3
y = -1
back into 1st :
2 - 2 - z =2
-z = 2
z = -2
so (2,-1,-2)
To solve the given system of equations, you can use the method of elimination or substitution. Let's solve it using the method of elimination:
Step 1: Multiply the second equation by 2:
The given system of equations becomes:
1. x + 2y - z = 2
2. -6x + 2y + 2z = -18
3. x + y - 2z = 5
Step 2: Add equation 1 to equation 3:
(x + 2y - z) + (x + y - 2z) = 2 + 5
2x + 3y - 3z = 7 (equation 4)
Step 3: Add equation 2 to equation 3:
(-6x + 2y + 2z) + (x + y - 2z) = -18 + 5
-5x + 3y = -13 (equation 5)
Step 4: Multiply equation 5 by 2:
-10x + 6y = -26 (equation 6)
Step 5: Add equation 4 to equation 6:
(2x + 3y - 3z) + (-10x + 6y) = 7 - 26
-8x + 9y - 3z = -19 (equation 7)
Step 6: Solve equations 7 and 5 simultaneously.
Multiply equation 5 by 3 and equation 7 by -1 to eliminate y:
-15x + 9y = -39 (equation 8)
8x - 9y + 3z = 19 (equation 9)
Step 7: Add equations 8 and 9:
(-15x + 9y) + (8x - 9y + 3z) = -39 + 19
-7x + 3z = -20 (equation 10)
Step 8: Rearrange equation 10 to solve for x:
-7x = -20 - 3z
x = (20 + 3z)/7
Step 9: Substitute the value of x in equation 4:
2(20 + 3z)/7 + 3y - 3z = 7
40 + 6z + 3y - 3z = 49
40 + 3z + 3y = 49 (equation 11)
Step 10: Rearrange equation 11 to solve for y:
3y = 49 - 40 - 3z
3y = 9 - 3z
y = (9 - 3z)/3
y = 3 - z
Step 11: Substitute the values of x and y in equation 2:
-6(20 + 3z)/7 + 2(3 - z) + 2z = -18
-120 - 18z / 7 + 6 - 2z + 2z = -18
-120 - 18z + 42 - 14z = -18 * 7
-30z = -186
z = -186 / -30
z = 31/5
Step 12: Substitute the values of z in equations 10 and 11, to solve for x and y:
x = (20 + 3(31/5))/7
x = -2
y = 3 - (31/5)
y = -2/5
Therefore, the solution to the given system of equations is (x, y, z) = (-2, -2/5, 31/5).
Your answer (-2, -11/3, -2) is close but not correct. The correct answer is (-2, -2/5, 31/5).