For (x+by=7) to be parallel and distinct to (2x+6y=18), which value of (b) is suitable?

Correct answer: (3)

The direct answer is A: \(b=3\). First simplify the second line: divide \(2x+6y=18\) by 2 to obtain \(x+3y=9\). The first line is \(x+by=7\). For the two lines to be parallel, the coefficients of x and y must have the same ratio, so b must be 3. With b=3, the equations become \(x+3y=7\) and \(x+3y=9\). They have identical left sides but different constants, so they cannot be the same line and never meet; hence they are parallel and distinct. Option A is correct. Option B, b=2, gives a different coefficient ratio and generally produces intersecting lines. Option C, b=6, also gives a different slope. Option D, b=1, likewise does not match the required direction. Notice that b=3 does not make the lines coincident because 7 is not 9. Exam cue: same x-y coefficient ratio but unequal constants means parallel distinct lines.