SOLUTION: Solve the system of equations using matrices and row of operations. x-y+z=-4 2x-3y+4z=-15 5x+y-2z=12

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Question 869345: Solve the system of equations using matrices and row of operations.
x-y+z=-4
2x-3y+4z=-15
5x+y-2z=12
Found 2 solutions by Edwin McCravy, richwmiller:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Here's one I did that's exactly like it. Go here and use it as a model
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http://www.algebra.com/algebra/homework/coordinate/Linear-systems.faq.question.361018.html

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
1x-1y+1z=-4
2x-3y+4z=-15
5x+1y-2z=12
GAUSS JORDAN METHOD
Add (-2 * row1) to row2
1 -1 1 -4
0 -1 2 -7
5 1 -2 12
Add (-5 * row1) to row3
1 -1 1 -4
0 -1 2 -7
0 6 -7 32
Divide row2 by -1
1 -1 1 -4
0 1 -2 7
0 6 -7 32
Add (-6 * row2) to row3
1 -1 1 -4
0 1 -2 7
0 0 5 -10
Divide row3 by 5
1 -1 1 -4
0 1 -2 7
0 0 1 -2
Add (2 * row3) to row2
1 -1 1 -4
0 1 0 3
0 0 1 -2
Add (-1 * row3) to row1
1 -1 0 -2
0 1 0 3
0 0 1 -2
Add (1 * row2) to row1
1 0 0 1
0 1 0 3
0 0 1 -2
CRAMER's rule
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables







First let A=%28matrix%283%2C3%2C1%2C-1%2C1%2C2%2C-3%2C4%2C5%2C1%2C-2%29%29. This is the matrix formed by the coefficients of the given system of equations.


Take note that the right hand values of the system are -4, -15, and 12 and they are highlighted here:




These values are important as they will be used to replace the columns of the matrix A.




Now let's calculate the the determinant of the matrix A to get abs%28A%29=-5. To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.



Notation note: abs%28A%29 denotes the determinant of the matrix A.



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Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bx%5D (since we're replacing the 'x' column so to speak).






Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=-5. Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.



To find the first solution, simply divide the determinant of A%5Bx%5D by the determinant of A to get: x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-5%29%2F%28-5%29=1



So the first solution is x=1




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We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C1%2C-1%2C1%2C2%2C-3%2C4%2C5%2C1%2C-2%29%29 again (this is the coefficient matrix).




Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5By%5D (since we're replacing the 'y' column in a way).






Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=-15.



To find the second solution, divide the determinant of A%5By%5D by the determinant of A to get: y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-15%29%2F%28-5%29=3



So the second solution is y=3




---------------------------------------------------------





Let's reset again by letting A=%28matrix%283%2C3%2C1%2C-1%2C1%2C2%2C-3%2C4%2C5%2C1%2C-2%29%29 which is the coefficient matrix.



Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bz%5D






Now compute the determinant of A%5Bz%5D to get abs%28A%5Bz%5D%29=10.



To find the third solution, divide the determinant of A%5Bz%5D by the determinant of A to get: z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%2810%29%2F%28-5%29=-2



So the third solution is z=-2




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Final Answer:




So the three solutions are x=1, y=3, and z=-2 giving the ordered triple (1, 3, -2)




Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.