Line intersect segment

Decide whether the line p : x + 2 y - 7 = 0 intersects the line segment given by points A[1, 1] and B[5, 3]

Correct answer:

q =  1

Step-by-step explanation:

p1​ : x + 2 y − 7 = 0 p2​: ax​+by​+c = 0 a=1  a⋅ 1+b⋅ 1+c=0 a⋅ 5+b⋅ 3+c=0 1⋅ 1+b⋅ 1+c=0 1⋅ 5+b⋅ 3+c=0  b+c=−1 3b+c=−5  Pivot:Row1↔Row2 3b+c=−5 b+c=−1  Row2−31​⋅ Row1→Row2 3b+c=−5 0.67c=0.67  c=0.666666670.66666667​=1 b=3−5−c​=3−5−1​=−2  b=−2 c=1  p2​: x−2y+1 = 0  x+2y−7=0 x−2y+1=0  x+2⋅ y−7=0 x−2⋅ y+1=0  x+2y=7 x−2y=−1  Row2−Row1→Row2 x+2y=7 −4y=−8  y=−4−8​=2 x=7−2y=7−2⋅ 2=3  x=3 y=2  [x;y] ∈ AB  1< x=3 < 5 q=1

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