Find the Variables

Problem

[[21],[−9]]=[[x],[y]]*([[0,1],[−3,1]]*[[x],[y]])

Solution

1. Perform matrix multiplication on the right side by multiplying the 2×2 matrix by the 2×1 vector.

[[0,1],[−3,1]]*[[x],[y]]=[[(0)*(x)+(1)*(y)],[(−3)*(x)+(1)*(y)]]

[[0,1],[−3,1]]*[[x],[y]]=[[y],[−3*x+y]]

2. Multiply the resulting vector by the scalar vector [[x],[y]] to set up the system of equations.

[[21],[−9]]=[[x(y)],[y*(−3*x+y)]]

[[21],[−9]]=[[x*y],[−3*x*y+y2]]

3. Set up the system of equations by equating corresponding components.

x*y=21

−3*x*y+y2=−9

4. Substitute the value of x*y from the first equation into the second equation.

−3*(21)+y2=−9

−63+y2=−9

5. Solve for y by isolating the variable.

y2=54

y=±√(,54)

y=±3√(,6)

6. Solve for x by substituting the values of y back into x*y=21

x=21/y

If *y=3√(,6)⇒x=21/(3√(,6))=7/√(,6)=(7√(,6))/6

If *y=−3√(,6)⇒x=21/(−3√(,6))=−7/√(,6)=−(7√(,6))/6

Final Answer

(x,y)=((7√(,6))/6,3√(,6)),(−(7√(,6))/6,−3√(,6))

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