Write as a Vector Equality y=-2/3x+4 , 2x+3y=12

Problem

y=−2/3*x+4,2*x+3*y=12

Solution

1. Identify that both equations represent the same line. Rearranging the first equation y=−2/3*x+4 into standard form gives 3*y=−2*x+12 which is equivalent to 2*x+3*y=12

2. Determine a point on the line by setting x=0 Substituting x=0 into y=−2/3*(0)+4 gives y=4 Thus, a position vector (r_0) is ([0],[4])

3. Find the direction vector by identifying the slope m=−2/3 A line with slope Δ(y)/Δ(x) has a direction vector v=([Δ(x)],[Δ(y)]) Here, v=([3],[−2])

4. Construct the vector equation of the line using the form r=(r_0)+t*v where t is a scalar parameter.

Final Answer

([x],[y])=([0],[4])+t*([3],[−2])

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