Find the coordinates of the point where
the line through the points (3,–4, –5) and (2,–3, 1) crosses the plane 2x + y + z = 7.

Given: line passing through the points (3,–4, –5) and (2,–3, 1), and a plane 2x + y + z = 7

To find: the coordinates of the point where
the given line crosses the given plane

The equation of line passing through two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is

So equation of line passing through two points A(3,–4, –5) and B(2,–3, 1) is

This is equal to constant, so

From this we get

Now let (x,y,z) be the coordinates of the point where the line crosses the plane 2x+y+z=7

Putting the x, y, z values from equation (i), (ii) and (iii), in the equation of the given plane, we get

2x+y+z=7

⇒ 2(3-k)+(k-4)+(6k-5)=7

⇒ 6-2k+k-4+6k-5=7

⇒ 5k-3=7

⇒ 5k=7+3

⇒ 5k=10

∴, k=2

Now substituting the value of k in equation (i), we get

x=3-k⇒ x=3-2⇒ x=1

Now substituting the value of k in equation (ii), we get

y=k-4⇒ y=2-4⇒ y=-2

Now substituting the value of k in equation (iii), we get

z=6k-5⇒ z=6(2)-5⇒ z=12-5⇒ z=7

Hence (1,-2,7) is the coordinates of the point where
the line through the points (3,–4, –5) and (2,–3, 1) crosses the plane 2x + y + z = 7.