# Consider the two lines L_1 : X=2t, y=1+2t, z=3t and L_2 : x=-8+4s , y=1+4s , z= 1+5s Find the point of intersection of the two lines.

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### 1 Answer

You need to set the expressions of x equal, such that:

`x = 2t = -8 + 4s`

You need to set the expressions of y equal, such that:

`y = 1 + 2t = 1 + 4s`

You need to set the expressions of z equal, such that:

`z = 3t = 1 + 5s`

You need to write t in terms of s, using the first equation, such that:

`t = -8/2 + (4s)/2 => t = -4 + 2s`

You need to replace `-4 + 2s` for t in the third equation, such that:

`3(-4 + 2s) = 1 + 5s => -12 + 6s = 1 + 5s => 6s - 5s = 12 + 1 => s = 13`

`t = -4 + 2*13 => t = 26 - 4 => t = 22`

You need to test the values `t = 22` and `s = 13` in the second equation, such that:

`1 + 2*22 = 1 + 4*13 => 45 != 53`

**Hence, since the parameters `t = 22` and` s = 13` does not check all three equations, hence, the given lines do not intersect each other.**