Solve Using a Matrix with Cramer's Rule y+x=-2 , 3+x=3-2y

Problem

{[y+x=−2],[3+x=3−2*y])

Solution

1. Rearrange the equations into standard form a*x+b*y=c

x+y=−2

x+2*y=0

2. Identify the coefficient matrix A the variable vector X and the constant vector B

A=[[1,1],[1,2]]

B=[[−2],[0]]

3. Calculate the determinant of the coefficient matrix D

D=(1)*(2)−(1)*(1)

D=1

4. Calculate the determinant (D_x) by replacing the first column of A with B

(D_x)=|[−2,1],[0,2]|

(D_x)=(−2)*(2)−(1)*(0)

(D_x)=−4

5. Calculate the determinant (D_y) by replacing the second column of A with B

(D_y)=|[1,−2],[1,0]|

(D_y)=(1)*(0)−(−2)*(1)

(D_y)=2

6. Solve for x and y using the formulas x=(D_x)/D and y=(D_y)/D

x=(−4)/1=−4

y=2/1=2

Final Answer

(x,y)=(−4,2)

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