SOLUTION: Please help with this! I need to solve this: -2x + 19y + 8z = 27 x-3y -2z = 5 2x-9y-5z = -35 Thank you so much.

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Please help with this! I need to solve this: -2x + 19y + 8z = 27 x-3y -2z = 5 2x-9y-5z = -35 Thank you so much.      Log On


   

Question 607167: Please help with this! I need to solve this:
-2x + 19y + 8z = 27
x-3y -2z = 5
2x-9y-5z = -35

Thank you so much.
Answer by jim_thompson5910(35256) 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







First let A=%28matrix%283%2C3%2C-2%2C19%2C8%2C1%2C-3%2C-2%2C2%2C-9%2C-5%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 27, 5, and -35 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=1. 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=524. 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=%28524%29%2F%281%29=524



So the first solution is x=524




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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%2C-2%2C19%2C8%2C1%2C-3%2C-2%2C2%2C-9%2C-5%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=-143.



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=%28-143%29%2F%281%29=-143



So the second solution is y=-143




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Let's reset again by letting A=%28matrix%283%2C3%2C-2%2C19%2C8%2C1%2C-3%2C-2%2C2%2C-9%2C-5%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=474.



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=%28474%29%2F%281%29=474



So the third solution is z=474




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Final Answer:




So the three solutions are x=524, y=-143, and z=474 giving the ordered triple (524, -143, 474)




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.