Use Cramer's rule to solve the following
4x+y-5z =11
To use Cramer's rule, we need to rewrite the system of equations in matrix form. The given equation can be written as:
4x + y - 5z = 11
Let's represent the coefficients and constants of this equation in matrix form:
[A][X] = [B]
Where [A] is the matrix of coefficients, [X] is the matrix of variables, and [B] is the matrix of constants.
For the given equation, [A] will be:
[4 1 -5]
[X] will be:
[x]
[y]
[z]
And [B] will be:
[11]
To use Cramer's rule, we need to find the determinant of the coefficient matrix [A] and the determinants of the matrices obtained by replacing each column of [A] with the matrix [B].
The determinant of [A] can be found as follows:
det[A] = 4(1)(-5) - (-5)(1) = -20 - (-5) = -20 + 5 = -15
Now, we need to find the determinants of the matrices obtained by replacing each column of [A] with [B].
The determinant of the matrix obtained by replacing the first column of [A] with [B] will be:
det[A1] = [11 1 -5] = 11(1)(-5) - (-5)(1) = -55 - (-5) = -55 + 5 = -50
Similarly, the determinants of the matrices obtained by replacing the second and third columns of [A] with [B] will be:
det[A2] = [4 11 -5] = 4(11)(-5) - (-5)(11) = -220 - (-55) = -220 + 55 = -165
det[A3] = [4 1 11] = 4(1)(11) - 11(4) = 44 - 44 = 0
Now, we can use Cramer's rule to find the values of x, y, and z:
x = det[A1] / det[A] = -50 / -15 = 10 / 3
y = det[A2] / det[A] = -165 / -15 = 11
z = det[A3] / det[A] = 0 / -15 = 0
Therefore, the solution to the equation 4x + y - 5z = 11 using Cramer's rule is x = 10/3, y = 11, and z = 0.