Introduced 81104

The * (asterisk) operation assigning one number to two pairs of numbers is introduced as follows:

(a, b)*(c, d) = ac+bd

We know that:
(x,2)*(-1, v) = -1 and (2,-1)*(u, v)=5 and (u, v)*(1,1)=-2

What is (1,2)*(x, y) equal to if y=3?

Correct answer:

w =  1

Step-by-step explanation:

y=3 (1,2)⋅ (x, y) = 1⋅ x+2⋅ y = x+6  (a, b)⋅ (c, d) = ac+bd = (c, d)⋅ (a,b)  −x+2v=−1 2u−v=5 u+v=−2  −x+2⋅ v=−1 2⋅ u−v=5 u+v=−2  2v−x=−1 2u−v=5 u+v=−2  Pivot:Row1↔Row2 2u−v=5 2v−x=−1 u+v=−2  Row3−21​⋅ Row1→Row3 2u−v=5 2v−x=−1 1.5v=−4.5  Row3−21.5​⋅ Row2→Row3 2u−v=5 2v−x=−1 0.75x=−3.75  x=0.75−3.75​=−5 v=2−1+x​=2−1−5​=−3 u=25+v​=25−3​=1  u=1 v=−3 x=−5  w=x+2⋅ y=(−5)+2⋅ 3=1

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