SOLUTION: help solving this sytem of equations? 5x + 2y - 5z = 31 2x - 4y + 2z = 6 4x - 5y + 4z = 6

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Question 929967: help solving this sytem of equations?
5x + 2y - 5z = 31
2x - 4y + 2z = 6
4x - 5y + 4z = 6
Found 2 solutions by TimothyLamb, ewatrrr:
Answer by TimothyLamb(4379) About Me  (Show Source):
You can put this solution on YOUR website!
5x + 2y - 5z = 31
2x - 4y + 2z = 6
4x - 5y + 4z = 6
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put the system of linear equations into standard form
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5x + 2y - 5z = 31
2x - 4y + 2z = 6
4x - 5y + 4z = 6
---
copy and paste the above standard form linear equations in to this solver:
https://sooeet.com/math/system-of-linear-equations-solver.php
---
solution:
x = 3
y = -2
z = -4
---
Solve and graph linear equations:
https://sooeet.com/math/linear-equation-solver.php
---
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---
Solve systems of linear equations up to 6-equations 6-variables:
https://sooeet.com/math/system-of-linear-equations-solver.php
---

Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
5x + 2y - 5z = 31
2x - 4y + 2z = 6
4x - 5y + 4z = 6
(3, -2, -4)
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables



system%285%2Ax%2B2%2Ay%2B-5%2Az=31%2C2%2Ax%2B-4%2Ay%2B2%2Az=6%2C4%2Ax%2B-5%2Ay%2B4%2Az=6%29



First let A=%28matrix%283%2C3%2C5%2C2%2C-5%2C2%2C-4%2C2%2C4%2C-5%2C4%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 31, 6, and 6 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=-60. 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=-180. 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=%28-180%29%2F%28-60%29=3



So the first solution is x=3




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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%2C5%2C2%2C-5%2C2%2C-4%2C2%2C4%2C-5%2C4%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=120.



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=%28120%29%2F%28-60%29=-2



So the second solution is y=-2




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Let's reset again by letting A=%28matrix%283%2C3%2C5%2C2%2C-5%2C2%2C-4%2C2%2C4%2C-5%2C4%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=240.



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=%28240%29%2F%28-60%29=-4



So the third solution is z=-4




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Final Answer:




So the three solutions are x=3, y=-2, and z=-4 giving the ordered triple (3, -2, -4)




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.