SOLUTION: Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. x - y + 2z = 6 4x + z = 1 x + 4y + z = -15

Algebra ->  Matrices-and-determiminant -> SOLUTION: Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. x - y + 2z = 6 4x + z = 1 x + 4y + z = -15      Log On


   

Question 385407: Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
x - y + 2z = 6
4x + z = 1
x + 4y + z = -15
Answer by Jk22(389) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Finding the Determinant of a 3x3 Matrix

If you have the general 3x3 matrix:

%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29

the determinant is:

Which further breaks down to:



Note: abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29, abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29 and abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 are determinants themselves.
If you need help finding the determinant of 2x2 matrices (which is required to find the determinant of 3x3 matrices), check out this solver

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From the matrix %28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29, we can see that a=1, b=-1, c=2, d=4, e=0, f=1, g=1, h=4, and i=1

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=-4--3%2B32 Multiply

abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=31 Combine like terms.


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Answer:

So abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=31, which means that the determinant of the matrix %28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29 is 31


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



system%281%2Ax%2B-1%2Ay%2B2%2Az=6%2C4%2Ax%2B0%2Ay%2B1%2Az=1%2C1%2Ax%2B4%2Ay%2B1%2Az=-15%29



First let A=%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%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 6, 1, and -15 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=31. 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=0. 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=%280%29%2F%2831%29=0



So the first solution is x=0




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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%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%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=-124.



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=%28-124%29%2F%2831%29=-4



So the second solution is y=-4




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Let's reset again by letting A=%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%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=31.



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=%2831%29%2F%2831%29=1



So the third solution is z=1




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Final Answer:




So the three solutions are x=0, y=-4, and z=1 giving the ordered triple (0, -4, 1)




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.