SOLUTION: hello how would i solve this linear system using the cramers rule? 4x-7y-7z= -26 3x-7y-9z=-26 -9x-5y-2z= -51 this is what i have done but it seems wrong i get that d = -

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: hello how would i solve this linear system using the cramers rule? 4x-7y-7z= -26 3x-7y-9z=-26 -9x-5y-2z= -51 this is what i have done but it seems wrong i get that d = -      Log On


   

Question 278155: hello
how would i solve this linear system using the cramers rule?
4x-7y-7z= -26
3x-7y-9z=-26
-9x-5y-2z= -51
this is what i have done but it seems wrong
i get that d = -145
dx = -1939/ -145
dy = 1175/-145
dz = -1428/-145
this answers seem to work only with the first equation.

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







First let A=%28matrix%283%2C3%2C4%2C-7%2C-7%2C3%2C-7%2C-9%2C-9%2C-5%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 -26, -26, and -51 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=-187. 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=-454. 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=%28-454%29%2F%28-187%29=454%2F187



So the first solution is x=454%2F187




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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%2C4%2C-7%2C-7%2C3%2C-7%2C-9%2C-9%2C-5%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=-1181.



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=%28-1181%29%2F%28-187%29=1181%2F187



So the second solution is y=1181%2F187




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





Let's reset again by letting A=%28matrix%283%2C3%2C4%2C-7%2C-7%2C3%2C-7%2C-9%2C-9%2C-5%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=227.



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=%28227%29%2F%28-187%29=-227%2F187



So the third solution is z=-227%2F187




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Final Answer:




So the three solutions are x=454%2F187, y=1181%2F187, and z=-227%2F187 giving the ordered triple (454/187, 1181/187, -227/187)




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.