Show that the lines x+13=y+35=z+57 and x+1=y+2=z+3 intersect. Also find their point of intersection.

Let x+13=y+35=z+57=p, x+1=y+2=z+3=q
General points on the lines are (3p-2, 5p-4, 7p-6) and (q-1, q-2, q-3)
If the lines intersect, then
3p-2 = q-1, 5p-4 = q-2, 7p-6 = q-3
or, 3p - q = 1 — (1)
5p - q = 2 — (2)
7p - q = 3 — (3)
Solving equation (1) and (2), we get
p = 1/2, q = -1/2
Putting the values of p and q in equation (3)
7(1/2) - (-1/2) = 4 ≠ 3
Therefore, lines do not intersect.
Let's use another method:
Let x+13=y+35=z+57=λ
Then x = λ-13, y = 5λ-18, z = 7λ-40
Let x+1=y+2=z+3=μ
Then x = μ-1, y = μ-2, z = μ-3
If they intersect, then
λ-13 = μ-1
5λ-18 = μ-2
7λ-40 = μ-3
Solving the first two equations,
λ-12 = μ
5λ-16 = μ
Then 5λ-16 = λ-12
4λ = 4
λ = 1
μ = -11
Substituting in the third equation:
7(1) - 40 = -11 -3
-33 = -14 (False)
Therefore, the lines do not intersect.