SOLUTION: Solve the system of equations with Cramers Rule 2x+3y+z=4 x-2y-4=0 3x-5z=-9

Algebra ->  Matrices-and-determiminant -> SOLUTION: Solve the system of equations with Cramers Rule 2x+3y+z=4 x-2y-4=0 3x-5z=-9      Log On


   

Question 862760: Solve the system of equations with Cramers Rule
2x+3y+z=4
x-2y-4=0
3x-5z=-9
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

x=118/41, y=-23/41, and z=-3/41 




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




  

  

  system%282%2Ax%2B3%2Ay%2B1%2Az=4%2C1%2Ax%2B-2%2Ay%2B0%2Az=4%2C3%2Ax%2B0%2Ay%2B-5%2Az=9%29

  

  

  

  First let A=%28matrix%283%2C3%2C2%2C3%2C1%2C1%2C-2%2C0%2C3%2C0%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 4, 4, and 9 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=41. 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=118. 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=%28118%29%2F%2841%29=118%2F41

  

  

  

  So the first solution is x=118%2F41

  

  

  

  

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

  

  

  We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C2%2C3%2C1%2C1%2C-2%2C0%2C3%2C0%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=-23.

  

  

  

  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=%28-23%29%2F%2841%29=-23%2F41

  

  

  

  So the second solution is y=-23%2F41

  

  

  

  

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

  

  

  

  

  

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

  

  

  

  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=%28-3%29%2F%2841%29=-3%2F41

  

  

  

  So the third solution is z=-3%2F41

  

  

  

  

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

  

  Final Answer:

  

  

  

  

  So the three solutions are x=118%2F41, y=-23%2F41, and z=-3%2F41 giving the ordered triple (118/41, -23/41, -3/41)

  

  

  

  

  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.