Answer:
solution: (x, y) = (-6, -10)
(see below for the "work")
Step-by-step explanation:
You are asked for the solution following the steps of evaluating Cramer's Rule.
System of equations in matrix form
Subtracting 2y from the second equation puts it in standard form, so the system is ...
Then the matrix equation is ...
[tex]\mathbf{AX}=\mathbf{B}\\\\\left[\begin{array}{cc}2&-1\\1&-2\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] =\left[\begin{array}{c}-2\\14\end{array}\right][/tex]
Determinant
The system determinant (Δ) is the determinant of the matrix of the coefficients of the variables.
[tex]\left|\begin{array}{cc}2&-1\\1&-2\end{array}\right|=(2)(-2)-(1)(-1)=-4+1=\boxed{-3}[/tex]
Y-determinant
The y-determinant (Δy) is the determinant of the matrix with the y-coefficients replaced by the constants (B).
[tex]\left|\begin{array}{cc}2&-2\\1&14\end{array}\right|=(2)(14)-(1)(-2)=28+2=\boxed{30}[/tex]
X-determinant
The x-determinant (Δx) is the determinant of the matrix with the x-coefficients replaced by the constants (B).
[tex]\left|\begin{array}{cc}-2&-1\\14&-2\end{array}\right|=(-2)(-2)-(14)(-1)=4+14=\boxed{18}[/tex]
Solution
The solution is the ratio of the appropriate determinants:
x = Δx/Δ = 18/-3 = -6
y = Δy/Δ = 30/-3 = -10
The solution is (x, y) = (-6, -10).
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The attachment shows a matching graphical solution.
