solve the system
2x-y+z=3
x+3y-2x=11
3x-2y+4z=1
double the first equation, then add that to the second to get
5x+y = 17
double the second, then add that to the third to get
5x + 4y = 23
now subtract those last two equations to get
3y = 6
y = 2
put that into 5x + y = 17 for
5x = 15
x = 3
now in the first
6 - 2 + z = 3
z = -1
I assumed your second equation had -2z at the end. typo ??
Thanks, yes, typo
I just finished solving it a long way, I solved the first two, then the second two,,etc etc. Your way seems much easier, I did get the same answers, yeah!!!! But, will your way always work?
To solve the given system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method:
Step 1: Let's multiply the second equation by 2 to align the coefficients of 'x':
2(x + 3y) - 2(2x) = 2(11)
2x + 6y - 4x = 22
-2x + 6y = 22
Step 2: Now, let's add the first equation and the equation obtained in Step 1:
(2x - y + z) + (-2x + 6y) = 3 + 22
- y + 7y + z = 25
6y + z = 25 (Equation A)
Step 3: Multiply the first equation by 3 to align the coefficients of 'x':
3(2x - y + z) = 3(3)
6x - 3y + 3z = 9
Step 4: Multiply the third equation by 2 to align the coefficients of 'z':
2(3x - 2y + 4z) = 2(1)
6x - 4y + 8z = 2
Step 5: Add the equations obtained in Step 3 and Step 4:
(6x - 3y + 3z) + (6x - 4y + 8z) = 9 + 2
12x - 7y + 11z = 11 (Equation B)
Step 6: Now, we can solve Equations A and B simultaneously. We have:
6y + z = 25 (Equation A)
12x - 7y + 11z = 11 (Equation B)
To find 'y', we can solve Equation A as follows:
Step 7: Multiply Equation A by -11 and add it to Equation B:
(-6y - z) + (12x - 7y + 11z) = -275 + 11
12x - 13y + 10z = -264 (Equation C)
Step 8: Multiply Equation A by 12 and add it to Equation B:
(72y + 12z) + (12x - 7y + 11z) = 12(25) + 11
12x + 65y + 23z = 311 (Equation D)
Step 9: Solve Equations C and D simultaneously:
12x - 13y + 10z = -264 (Equation C)
12x + 65y + 23z = 311 (Equation D)
Now, you can solve the system using the method of your choice (substitution, elimination, or matrices) to find the values of 'x', 'y', and 'z'.