Answer:
The determinant of A (the main matrix) is -3; the determinant of y is 30; the determinant of x is 18; the solution to the system is (-6, -10)
Step-by-step explanation:
Set up the matrix to find the determinant of the main matrix. Â Find the determinant by multiplying the numbers on the major axis and subtract from that the multiplication of the numbers on the minor axis:
[tex]\left[\begin{array}{ccc}2&-1&\\1&-2\\\end{array}\right][/tex]
Find the determinant by multiplication:
(2×-2)-(1×-1)= -3
To find the determinant of y, replace the second column with the solutions to have a matrix that looks like this:
[tex]\left[\begin{array}{ccc}2&-2\\1&14\\\end{array}\right][/tex]
To find the determinant of that matrix by multiplication:
(2×14)- (1× -2) = 30
Lastly, find the determinant of x by replacing the first column with the solutions. Â That matrix will look like this:
[tex]\left[\begin{array}{ccc}-2&-1\\14&-2\\\end{array}\right][/tex]
Find the determinant of x by multiplication:
(-2 × -2) - (14 × -1) = 18
Now we want Cramer's Rule that tells us if we divide the determinant of [tex]A_{x}[/tex]
by the determinant of A, we will find the value of x:
[tex]\frac{A_{x} }{A}=\frac{18}{-3} Â =-6[/tex]
and the same for y:
[tex]\frac{A_{y} }{A}=\frac{30}{-3}=-10[/tex]
So the solution to the system is (-6, -10)
