Solve Using an Inverse Matrix x-3y=5 , 5x-25=15y

Problem

{[x−3*y=5],[5*x−25=15*y])

Solution

1. Rearrange the second equation into standard form a*x+b*y=c by adding 25 to both sides and subtracting 15*y from both sides.

x−3*y=5

5*x−15*y=25

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

A=[[1,−3],[5,−15]]

X=[[x],[y]]

B=[[5],[25]]

3. Calculate the determinant of matrix A denoted as det(A) using the formula a*d−b*c

det(A)=(1)*(−15)−(−3)*(5)

det(A)=−15−(−15)

det(A)=0

4. Determine the solvability of the system. Since the determinant is 0 the matrix A is singular and does not have an inverse.

det(A)=0⇒A(−1)* does not exist

5. Analyze the equations to check for infinite solutions or no solution. Divide the second equation 5*x−15*y=25 by 5

(5*x−15*y)/5=25/5

x−3*y=5

6. Conclude that because both equations are identical, the system is dependent and has infinitely many solutions.

x=3*y+5

Final Answer

Infinitely many solutions of the form *(3*y+5,y)

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