Solve Using an Inverse Matrix x=-3y+39 , 3x+4y=67

Problem

{[x=−3*y+39],[3*x+4*y=67])

Solution

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

x+3*y=39

3*x+4*y=67

2. Write the system as a matrix equation A*X=B

A=[[1,3],[3,4]]

X=[[x],[y]]

B=[[39],[67]]

3. Calculate the determinant of matrix A denoted as det(A)

det(A)=(1)*(4)−(3)*(3)

det(A)=4−9=−5

4. Find the inverse matrix A(−1) using the formula A(−1)=1/det(A)*[[d,−b],[−c,a]]

A(−1)=1/(−5)*[[4,−3],[−3,1]]

A(−1)=[[−0.8,0.6],[0.6,−0.2]]

5. Solve for X by multiplying A(−1) and B

X=A(−1)*B

X=[[−0.8,0.6],[0.6,−0.2]]*[[39],[67]]

6. Compute the values for x and y

x=(−0.8)*(39)+(0.6)*(67)=−31.2+40.2=9

y=(0.6)*(39)+(−0.2)*(67)=23.4−13.4=10

Final Answer

[[x],[y]]=[[9],[10]]

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