# Find the intersection of the following two lines. L_1: (x-5)/2= (y+3)/1 = (z+4)/5 L_2: (x-12)/1= (y-11)/-3 = (z-15)/2

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to convert the given symmetrical form of `L_1` into parametric form, such that:

`(x - 5)/2 = t => x = 5 + 2t`

`y + 3 = t =>y = -3 + t`

`(z + 4)/5 = t => z = -4 + 5t`

You need to convert the given symmetrical form of `L_2` into parametric form, such that:

`x - 12 = s => x = 12 + s`

`(y - 11)/(-3)= s => y = 11 - 3s`

`(z - 15)/2 = s=> z = 15 + 2s`

You need to equate the parametric forms of x,y and z, such that:

`{(5 + 2t = 12 + s),(-3 + t = 11 - 3s),(-4 + 5t = 15 + 2s):}`

`{(2t - s = 12 - 5),(t + 4s = 11 + 3),(5t - 2s = 15 + 4):}`

`{(2t - s = 7),(t + 4s = 14),(5t - 2s = 19):}`

`{(s = 2t - 7),(t + 4(2t - 7) = 14),(5t - 2(2t - 7) = 19):}`

`{(s = 2t - 7),(t + 8t - 28 = 14),(5t - 4t + 14 = 19):}`

`{(s = 2t - 7),(t = 42/9),(t = 33):}`

You need to notice that using the substitution` s = 2t - 7` yields two different values of t, hence, the given lines `L_1,L_2` do not intersect each other.