Show the two lines (x/2-1/2)=(y/3-2/3)=(z/4-3/4) and (x/5-4/5)=z intersect.find the point of intersection of these lines
subbing the 2nd into the first:
x/2 - 1/2 = z/4 - 3/4
x/2 - 1/2 = (1/4)(x/5-4/5) - 3/4
times 4
2x - 2 = (x/5 - 4/5) - 3
times 5
10x - 10 = x - 4 - 15
9x = -9
x = -1
into z = x/5 - 4/4 = -1/5 - 4/5 = -1
z = -1
lastly
y/3 - 2/3 = z/4 - 3/4
y/3 - 2/3 = -1/4-3/4
y/3 - 2/3 = -1
times 3
y - 2 = -3
y = -1
they intersect at (-1, -1, -1)
To find the point of intersection between two lines, we need to solve the equations representing each line simultaneously. Let's start by writing the equations of the two lines:
Line 1: x/2 - 1/2 = y/3 - 2/3 = z/4 - 3/4
Line 2: x/5 - 4/5 = z
We can rewrite Line 1 as three separate equations:
1) x/2 - 1/2 = y/3 - 2/3
2) x/2 - 1/2 = z/4 - 3/4
3) y/3 - 2/3 = z/4 - 3/4
Now, let's solve Line 2 for x in terms of z:
x/5 - 4/5 = z
x/5 = z + 4/5
x = 5z + 4/5
Now, substitute this expression for x in the three equations from Line 1:
1) (5z + 4/5)/2 - 1/2 = y/3 - 2/3
2) (5z + 4/5)/2 - 1/2 = z/4 - 3/4
3) y/3 - 2/3 = z/4 - 3/4
Simplifying each equation, we get:
1) (5z + 4/5)/2 - 1/2 = y/3 - 2/3
5z + 4/5 - 1/2 = 2y/6 - 4/6
5z + 4/5 - 1/2 = 2y/6 - 4/6
5z + 4/5 - 2/5 = 2y/6 - 2/3
5z + 12/15 - 6/15 = 10y/30 - 10/30
5z + 6/15 = 10y/30
5z + 6/15 = y/3
5z + 2/5 = y/3
15z + 6/5 = y
2) (5z + 4/5)/2 - 1/2 = z/4 - 3/4
5z + 4/5 - 1/2 = z/4 - 3/4
5z + 4/5 - 2/5 = z/4 - 3/4
5z + 12/15 - 6/15 = z/4 - 12/15
5z + 6/15 = z/4 - 12/15
5z + 6/15 + 12/15 = z/4
5z + 18/15 = z/4
20z + 18 = 15z
20z - 15z = -18
5z = -18
z = -18/5
We can substitute the value of z back into the expression we found for y above:
y = 15z + 6/5
y = 15(-18/5) + 6/5
y = -54 + 6/5
y = -54 + 30/5
y = -54 + 6
y = -48
Finally, substitute the values of y and z into the expression we found for x above:
x = 5z + 4/5
x = 5(-18/5) + 4/5
x = -18 + 4/5
x = -18 + 4/5
x = -18 + 4/5
x = -18 + 4/5
x = -18 + 0.8
x = -17.2
So, the point of intersection of the two lines is (-17.2, -48, -18/5).