Question

In Exercises 31 and 32 , use a computer algebra system to graph the pair of intersecting lines and find the point of intersection. $$ \begin{array}{l} x=2 t+3, y=5 t-2, z=-t+1 \\ x=-2 s+7, y=s+8, z=2 s-1 \end{array} $$

Short answer

The lines intersect at the point (7,8,-1).

Step-by-step solution

Graph the Lines

Use a computer algebra system (CAS) to graph the lines to visualize the problem. The lines are given by their parametric equations as \(x=2 t+3, y=5 t-2, z=-t+1\) and \(x=-2 s+7, y=s+8, z=2 s-1\).

Set Up System of Equations

To find the point of intersection, the x, y, and z components from the parametric equations of the lines should be set equal to each other. Namely, \(2t+3 = -2s+7, 5t-2 = s+8, -t+1=2s-1\).

Solve System of Equations

The simultaneous equations can be solved using substitution or elimination method. By rearranging and solving, one can find that \(s=1\) and \(t=2\).

Find Intersection Point

Substitute the solved values \(s=1\) and \(t=2\) back into the parametric equations to find the exact intersection point. This results in \(x=7, y=8, z=-1\).