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.Com
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)   (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:



the determinant is:

Which further breaks down to:



Note: , and 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

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


From the matrix , we can see that , , , , , , , , and

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

Multiply

Combine like terms.


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


Answer:

So , which means that the determinant of the matrix is 31


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







First let . 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 , , and 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 . 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: 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 (since we're replacing the 'x' column so to speak).






Now compute the determinant of to get . 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 by the determinant of to get:



So the first solution is




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


We'll follow the same basic idea to find the other two solutions. Let's reset by letting 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 (since we're replacing the 'y' column in a way).






Now compute the determinant of to get .



To find the second solution, divide the determinant of by the determinant of to get:



So the second solution is




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





Let's reset again by letting 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






Now compute the determinant of to get .



To find the third solution, divide the determinant of by the determinant of to get:



So the third solution is




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

Final Answer:




So the three solutions are , , and 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.
RELATED QUESTIONS

Use Cramer's rule to solve the system of equations. If D = 0, use another method to... (answered by jim_thompson5910)
x+y=11 Use Cramer's rule to solve the x-y=5 following system of equations. If D=0, use... (answered by Edwin McCravy)
How can I use Cramer’s rule to solve for y in the following system of equations. x +... (answered by Alan3354)
I DONT UNDERSTAND THIS PLEASE HELP..! Use Cramer’s rule to solve for y in the... (answered by tlwright)
When using Cramer’s Rule: x= (33 5 51 7)/ (3 5 (answered by richwmiller)
When using Cramer’s Rule: x= (33 5) 51 7/ (3 5) (answered by Fombitz)
Use the Cramer’s rule to solve the following system of linear equations; x - y + z = 8, (answered by Edwin McCravy)
Use Cramer's rule to solve the following system of equations. If D = 0, use another... (answered by Alan3354)
Use the Gauss-Jordan method to solve the system of equations. x-y=2 y-z=0 x+z=6... (answered by jim_thompson5910)