Question

Solve the system of equations.
–5x + 3y + 2z = 19
x + 4y + 3z = –27
15x – 9y – 6z = –55

Answer 1

Do you want to know how it's done, or the answers?

Answer 2

both will be great if you have the time...

Answer 3

Does it have to be done by Cramer's rule, or by substitution/elimination, or by matrix reduction?
Does any of these methods sound familiar to you?

Answer 4

all of them sounds familiar so any can be used

Answer 5

Have you worked with determinants, and be able to work out a 3x3 determinant?

Answer 6

yes! Can I do this problem that way?

Please tell me how.

Answer 7

ok, so we'll solve by Cramer's rule.

First, we'll reduce the system to matrix form, namely
AX=B
where A is the 3x3 matrix of coefficients of the system,
X is the solution vector, and
B is the vector on the right side of the equations.
c1 is the first column of the matrix, i.e.
[-5,1,15],
c2 is [3,4,-9], and c3 is [2,3,-6].
B is [19,-27,-55].
ok so far?

Answer 8

I just found out that the matrix is not invertible, because the determinant is zero.
Would you like to check the question to make sure that the numbers are correct?
If they are correct, then Cramer's rule cannot find a solution since the matrix is not invertible.
However, since the equations are linearly dependent, there are infinite solutions, if you have already done that in your course.

Answer 9

The solution by row reduction is
x=-z/23, y=-17z/23
where z can be any real number, therefore an infinite number of solutions.

Answer 10

Thank you!

Answer 11

You're welcome!