Write as a Vector Equality x/5=y/4 , y/3=x/3-1

Problem

x/5=y/4,y/3=x/3−1

Solution

1. Rearrange the first equation into standard form a*x+b*y=c by multiplying by the least common multiple, 20.

x/5−y/4=0

4*x−5*y=0

2. Rearrange the second equation into standard form by multiplying by 3 and moving variables to one side.

y=x−3

−x+y=−3

3. Identify the coefficients of x and y from both equations to form the column vectors.

Equation 1: *4*x−5*y=0

Equation 2: −1*x+1*y=−3

4. Express the system as a linear combination of vectors by grouping the x terms, the y terms, and the constants.

x*([4],[−1])+y*([−5],[1])=([0],[−3])

Final Answer

x*([4],[−1])+y*([−5],[1])=([0],[−3])

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