SOLUTION: Please help use Cramer's rule to solve the linear system. ( 3x-2y-z=3 x+2y-5z=-31 -7x+y+z=25

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Question 618812: Please help use Cramer's rule to solve the linear system.
( 3x-2y-z=3
x+2y-5z=-31
-7x+y+z=25
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi,

Please help use Cramer's rule to solve the linear system.

 3x-2y-z=3

  x+2y-5z=-31

  -7x+y+z=25 

 x= -145/31 y = -288/31 z =  48/31)






 
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables




  

  

  

  

  

  

  First let A=%28matrix%283%2C3%2C3%2C-2%2C-1%2C1%2C2%2C-5%2C-7%2C1%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 3, -31, and 25 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=-62. 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.

  

  

  

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

  

  

  

  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=290. 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=%28290%29%2F%28-62%29=-145%2F31

  

  

  

  So the first solution is x=-145%2F31

  

  

  

  

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

  

  

  We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C3%2C-2%2C-1%2C1%2C2%2C-5%2C-7%2C1%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=576.

  

  

  

  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=%28576%29%2F%28-62%29=-288%2F31

  

  

  

  So the second solution is y=-288%2F31

  

  

  

  

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

  

  

  

  

  

  Let's reset again by letting A=%28matrix%283%2C3%2C3%2C-2%2C-1%2C1%2C2%2C-5%2C-7%2C1%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=-96.

  

  

  

  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=%28-96%29%2F%28-62%29=48%2F31

  

  

  

  So the third solution is z=48%2F31

  

  

  

  

  ====================================================================================

  

  Final Answer:

  

  

  

  

  So the three solutions are x=-145%2F31, y=-288%2F31, and z=48%2F31 giving the ordered triple (-145/31, -288/31, 48/31)

  

  

  

  

  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.