MHB Finding the Coordinates of Point D

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To find the coordinates of point D, the equation AD is established as a combination of vectors BC, AB, and AC. The vectors are calculated as AB = (4,-2), BC = (-3,-9), and AC = (1,-11). By equating the components of the vector AD with the derived expressions, a system of equations is formed to solve for the variables x and y. After solving these equations, the coordinates of point D can be determined. The final coordinates of D are derived from the values of x and y obtained from the system.
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Points A ,B and C have coordinates A(1,3) B(5,1) and C(2,-8).
point D is such that the vector AD = Vector BC + (2x) vector AB + (3y) vectorAC = vector AB + (2x) vectorAC + (3y) vector BC
find coordinates of D
 
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AB = (4,-2), BC = (-3,-9), AC = (1,-11)

let D = (m,n) $\implies$ AD = (m-1,n-3)

AD = (-3,-9) + 2x(4,-2) + 3y(1,-11) = (4,-2) + 2x(1,-11) + 3y(-3,-9)

AD = (-3+8x+3y , -9-4x-33y) = (4+2x-9y , -2-22x-27y)

-3+8x+3y = 4+2x-9y
-9-4x-33y = -2-22x-27y

solve the system for x and y, then determine the coordinates of D
 

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