SOLUTION: -3x + y + 11z = -410 -4x - 6y + 5z = -214 -2x - 2y + 6z = -244 I need to solve this and show work any help would be greatly appreciated !!

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: -3x + y + 11z = -410 -4x - 6y + 5z = -214 -2x - 2y + 6z = -244 I need to solve this and show work any help would be greatly appreciated !!      Log On


   

Question 429953: -3x + y + 11z = -410
-4x - 6y + 5z = -214
-2x - 2y + 6z = -244
I need to solve this and show work any help would be greatly appreciated !!
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

-3x + y + 11z = -410

-4x - 6y + 5z = -214

-2x - 2y + 6z = -244




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




  

  

  

  

  

  

  First let A=%28matrix%283%2C3%2C-3%2C1%2C11%2C-4%2C-6%2C5%2C-2%2C-2%2C6%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 -410, -214, and -244 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=48. 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=-672. 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=%28-672%29%2F%2848%29=-14

  

  

  

  So the first solution is x=-14

  

  

  

  

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

  

  

  We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C-3%2C1%2C11%2C-4%2C-6%2C5%2C-2%2C-2%2C6%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=480.

  

  

  

  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=%28480%29%2F%2848%29=10

  

  

  

  So the second solution is y=10

  

  

  

  

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

  

  

  

  

  

  Let's reset again by letting A=%28matrix%283%2C3%2C-3%2C1%2C11%2C-4%2C-6%2C5%2C-2%2C-2%2C6%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=-2016.

  

  

  

  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=%28-2016%29%2F%2848%29=-42

  

  

  

  So the third solution is z=-42

  

  

  

  

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

  

  Final Answer:

  

  

  

  

  So the three solutions are x=-14, y=10, and z=-42 giving the ordered triple (-14, 10, -42)

  

  

  

  

  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.