SOLUTION: x+y-z=6 2x-y+3z=1 x+3y+z=8 solve by cramer's rule

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Question 652143: x+y-z=6
2x-y+3z=1
x+3y+z=8
solve by cramer's rule
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



system%281%2Ax%2B1%2Ay%2B-1%2Az=6%2C2%2Ax%2B-1%2Ay%2B3%2Az=1%2C1%2Ax%2B3%2Ay%2B1%2Az=8%29



First let A=%28matrix%283%2C3%2C1%2C1%2C-1%2C2%2C-1%2C3%2C1%2C3%2C1%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 6, 1, and 8 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=-16. 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=-48. 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=%28-48%29%2F%28-16%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%2C1%2C1%2C-1%2C2%2C-1%2C3%2C1%2C3%2C1%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=-32.



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=%28-32%29%2F%28-16%29=2



So the second solution is y=2




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Let's reset again by letting A=%28matrix%283%2C3%2C1%2C1%2C-1%2C2%2C-1%2C3%2C1%2C3%2C1%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=16.



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=%2816%29%2F%28-16%29=-1



So the third solution is z=-1




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Final Answer:




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




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.