SOLUTION: Can you help me solve the system of equations using matrices: x-2y+3z=7 2x+y+z=5 -3x+2y-2z=-10

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Question 380278: Can you help me solve the system of equations using matrices: x-2y+3z=7 2x+y+z=5 -3x+2y-2z=-10
Answer by Jk22(389) 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



system%281%2Ax%2B-2%2Ay%2B3%2Az=7%2C2%2Ax%2B1%2Ay%2B1%2Az=5%2C-3%2Ax%2B2%2Ay%2B-2%2Az=-10%29



First let A=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%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 7, 5, and -10 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=15. 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=32. 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=%2832%29%2F%2815%29=32%2F15



So the first solution is x=32%2F15




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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-2%2C3%2C2%2C1%2C1%2C-3%2C2%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=-8.



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=%28-8%29%2F%2815%29=-8%2F15



So the second solution is y=-8%2F15




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Let's reset again by letting A=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%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=19.



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=%2819%29%2F%2815%29=19%2F15



So the third solution is z=19%2F15




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Final Answer:




So the three solutions are x=32%2F15, y=-8%2F15, and z=19%2F15 giving the ordered triple (32/15, -8/15, 19/15)




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.




det(A) is
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

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


From the matrix %28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29, we can see that a=1, b=-2, c=3, d=2, e=1, f=1, g=-3, h=2, and i=-2

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=-4-2%2B21 Multiply

abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=15 Combine like terms.


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


Answer:

So abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=15, which means that the determinant of the matrix %28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29 is 15


det(Ax) is
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

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


From the matrix %28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29, we can see that a=7, b=-2, c=3, d=5, e=1, f=1, g=-10, h=2, and i=-2

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=-28-0%2B60 Multiply

abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=32 Combine like terms.


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


Answer:

So abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=32, which means that the determinant of the matrix %28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29 is 32



det(Ay) is
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

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


From the matrix %28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29, we can see that a=1, b=7, c=3, d=2, e=5, f=1, g=-3, h=-10, and i=-2

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=0--7%2B-15 Multiply

abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=-8 Combine like terms.


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


Answer:

So abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=-8, which means that the determinant of the matrix %28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29 is -8


det(Az) is
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

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


From the matrix %28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29, we can see that a=1, b=-2, c=7, d=2, e=1, f=5, g=-3, h=2, and i=-10

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=-20-10%2B49 Multiply

abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=19 Combine like terms.


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


Answer:

So abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=19, which means that the determinant of the matrix %28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29 is 19