Show that the lines x/1 = (y + 7)/3 = (z − 8)/(−2) and (x − 4)/2 = (y + 2)/(−1) = (z − 5)/1 intersect. Find their point of intersection.

1. 1 · Write what is given and what is to be found. Given: two lines in Cartesian form. To show: they meet. To find: the common point.
2. 2 · Draw the figure or sketch. Sketch two lines crossing at a point and label the point P.
3. 3 · Write the rule. The lines meet if one pair λ, μ makes their general points equal in all three coordinates.
4. 4 · Write the general points.
   • L₁: (λ, −7 + 3λ, 8 − 2λ)
   • L₂: (4 + 2μ, −2 − μ, 5 + μ)
5. 5 · Solve using x and y.
   • x: λ = 4 + 2μ
   • y: −7 + 3(4 + 2μ) = −2 − μ, so 5 + 6μ = −2 − μ
   • 7μ = −7, so μ = −1 and λ = 2
6. 6 · Check the z-coordinates. L₁: 8 − 2(2) = 4. L₂: 5 + (−1) = 4. Equal, so the lines intersect.
7. 7 · Check and box the answer. Point: (2, −7 + 6, 4) = (2, −1, 4). L₂ at μ = −1 also gives (2, −1, 4).

Scoring tip: Use two coordinates to find λ and μ, and the third to check. Without the check, you have not shown that the lines meet.