The following expression represents the value of which variable in the solution of the system of equations below?
|2 -1 1|
|1 2 6|
|3 -1 2|
___________
|2 -1 -2|
|1 2 3|
|3 -1 -1|
2X-Y+Z=-2
X+2Y+6Z=3
3X-Y+2Z=-1
A. Z
B. Y
C. X
D. NONE OF THE VARIABLES ARE REPRESENTED
You replace the third column in the NUMERATOR determinant with the right hand side, NOT THE DENOMINATOR to get the third variable, Z. Therefore you got 1/Z
Here is a link to Cramer's rule:
http://www.purplemath.com/modules/cramers.htm
To solve the system of equations, we can use the method of matrix inversion. The system of equations can be represented in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
First, let's represent the system of equations in matrix form:
|2 -1 1| |X| |-2|
|1 2 6| |Y| | 3|
|3 -1 2| * |Z| = |-1|
To find the variable matrix X, we need to find the inverse of the coefficient matrix A. In other words, we need to find A^(-1) such that A^(-1) * A = I, where I is the identity matrix.
Calculating the inverse of A, we get:
|4/23 3/23 -9/23|
|-10/23 5/23 17/23|
|13/46 -1/46 1/46|
Now, we can find the variable matrix X by multiplying both sides of the equation AX = B by A^(-1):
A^(-1) * AX = A^(-1) * B
X = A^(-1) * B
Calculating X, we get:
|X| |4/23 3/23 -9/23| |-2|
|Y| = |-10/23 5/23 17/23| * | 3|
|Z| |13/46 -1/46 1/46| |-1|
Simplifying the expression for X, we have:
X = (4/23)(-2) + (3/23)(3) + (-9/23)(-1)
X = -8/23 + 9/23 + 9/23
X = 10/23
Similarly, simplifying for Y and Z, we have:
Y = (-10/23)(-2) + (5/23)(3) + (17/23)(-1)
Y = 20/23 + 15/23 - 17/23
Y = 18/23
Z = (13/46)(-2) + (-1/46)(3) + (1/46)(-1)
Z = -26/46 - 3/46 - 1/46
Z = -30/46
Z = -15/23
Therefore, the values of the variables X, Y, and Z in the solution of the system of equations are:
X = 10/23
Y = 18/23
Z = -15/23
The expression |X| represents the value of the variable X in the solution. Therefore, the answer is C. X.