Write as a Vector Equality y=1/3x-2 , 3y=x-6

Problem

y=1/3*x−2,3*y=x−6

Solution

1. Identify the relationship between the two equations. By multiplying the first equation by 3 we get 3*y=x−6 which is identical to the second equation. This represents a single line in ℝ2

2. Express the line in general form a*x+b*y=c Subtracting 1/3*x from both sides of the first equation gives −1/3*x+y=−2 or x−3*y=6

3. Parameterize the variables by setting one variable in terms of a parameter t Let y=t

4. Solve for x using the parameter t From x−3*y=6 we substitute y=t to get x=6+3*t

5. Write the solution in vector form y=(x_0)+t*v The vector ([x],[y]) can be written as ([6+3*t],[t])

6. Separate the constant vector and the parameter vector to reach the final vector equality.

([x],[y])=([6],[0])+t*([3],[1])

Final Answer

([x],[y])=([6],[0])+t*([3],[1])

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