SOLUTION: I have to show my work and I only know how to do these on a calculator any help would be greatly appreciated!! -4x - 5y - 8z = 268 -10x + 7y + 6z = -786 -12x + 9y - 2z = -548

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: I have to show my work and I only know how to do these on a calculator any help would be greatly appreciated!! -4x - 5y - 8z = 268 -10x + 7y + 6z = -786 -12x + 9y - 2z = -548      Log On


   

Question 429955: I have to show my work and I only know how to do these on a calculator any help would be greatly appreciated!!
-4x - 5y - 8z = 268
-10x + 7y + 6z = -786
-12x + 9y - 2z = -548
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

-4x - 5y - 8z = 268

-10x + 7y + 6z = -786

-12x + 9y - 2z = -548




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




  

  

  

  

  

  

  First let A=%28matrix%283%2C3%2C-4%2C-5%2C-8%2C-10%2C7%2C6%2C-12%2C9%2C-2%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 268, -786, and -548 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=780. 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=31980. 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=%2831980%29%2F%28780%29=41

  

  

  

  So the first solution is x=41

  

  

  

  

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

  

  

  We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C-4%2C-5%2C-8%2C-10%2C7%2C6%2C-12%2C9%2C-2%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=-12480.

  

  

  

  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=%28-12480%29%2F%28780%29=-16

  

  

  

  So the second solution is y=-16

  

  

  

  

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

  

  

  

  

  

  Let's reset again by letting A=%28matrix%283%2C3%2C-4%2C-5%2C-8%2C-10%2C7%2C6%2C-12%2C9%2C-2%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=-34320.

  

  

  

  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=%28-34320%29%2F%28780%29=-44

  

  

  

  So the third solution is z=-44

  

  

  

  

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

  

  Final Answer:

  

  

  

  

  So the three solutions are x=41, y=-16, and z=-44 giving the ordered triple (41, -16, -44)

  

  

  

  

  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.