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

Problem

y=2*x+3,1/2*y=x+3/2

Solution

1. Analyze the given equations to determine if they represent the same line.

2. Multiply the second equation by 2 to compare it to the first.

2*(1/2*y)=2*(x+3/2)

y=2*x+3

3. Identify that both equations represent the same line, so we only need to parameterize one of them.

4. Set the independent variable x equal to a parameter t

x=t

5. Substitute x=t into the equation y=2*x+3 to find the expression for y

y=2*t+3

6. Write the variables as a column vector.

([x],[y])=([t],[2*t+3])

7. Decompose the vector into a constant vector and a vector containing the parameter t

([x],[y])=([0],[3])+([t],[2*t])

8. Factor out the parameter t to reach the final vector form.

([x],[y])=([0],[3])+t*([1],[2])

Final Answer

([x],[y])=([0],[3])+t*([1],[2])

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