SOLUTION: Solution for this problem using the matrices 3x-4y+2z=-32 2x+3y=8 y-3z=19

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Question 870749: Solution for this problem using the matrices
3x-4y+2z=-32
2x+3y=8
y-3z=19
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables



system%283%2Ax%2B-4%2Ay%2B2%2Az=-32%2C2%2Ax%2B3%2Ay%2B0%2Az=8%2C0%2Ax%2B1%2Ay%2B-3%2Az=19%29



First let A=%28matrix%283%2C3%2C3%2C-4%2C2%2C2%2C3%2C0%2C0%2C1%2C-3%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 -32, 8, and 19 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=-47. 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=94. 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=%2894%29%2F%28-47%29=-2



So the first solution is x=-2




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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%2C3%2C-4%2C2%2C2%2C3%2C0%2C0%2C1%2C-3%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=-188.



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=%28-188%29%2F%28-47%29=4



So the second solution is y=4




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Let's reset again by letting A=%28matrix%283%2C3%2C3%2C-4%2C2%2C2%2C3%2C0%2C0%2C1%2C-3%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=235.



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=%28235%29%2F%28-47%29=-5



So the third solution is z=-5




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Final Answer:




So the three solutions are x=-2, y=4, and z=-5 giving the ordered triple (-2, 4, -5)




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.