Write as a Vector Equality

Problem

(x+y)/7=(y+4)/5,(x−z)/5=(y−4)/2,(y−z)/3=(x+2)/10

Solution

1. Clear the denominators for each equation by multiplying by the least common multiple of the denominators.

5*(x+y)=7*(y+4)

2*(x−z)=5*(y−4)

10*(y−z)=3*(x+2)

2. Expand and rearrange each equation into the standard form a*x+b*y+c*z=d

5*x+5*y=7*y+28⇒5*x−2*y+0*z=28

2*x−2*z=5*y−20⇒2*x−5*y−2*z=−20

10*y−10*z=3*x+6⇒−3*x+10*y−10*z=6

3. Identify the coefficients of the variables x y and z to form the coefficient matrix A and the constant vector b

A=([5,−2,0],[2,−5,−2],[−3,10,−10])

b=([28],[−20],[6])

4. Express as a vector equality by writing the system in the form A*x=b

([5,−2,0],[2,−5,−2],[−3,10,−10])*([x],[y],[z])=([28],[−20],[6])

Final Answer

([5,−2,0],[2,−5,−2],[−3,10,−10])*([x],[y],[z])=([28],[−20],[6])

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