Question

Use a computer algebra system to graph the pair of intersecting lines and find the point of intersection. $$ \begin{array}{l} x=2 t-1, y=-4 t+10, z=t \\ x=-5 s-12, y=3 s+11, z=-2 s-4 \end{array} $$

Short answer

The point of intersection of the two lines are the solutions to the equations obtained from the steps and would be the final answer.

Step-by-step solution

Set up the system of equations

The two lines are given parametrically as: \(x=2t-1, y=-4t+10, z=t\) and \(x=-5s-12, y=3s+11, z=-2s-4\). We need to solve this system to find the point of intersection. This implies that \(2t-1 = -5s-12, -4t+10 = 3s+11, t = -2s-4\).

Isolate variables

Isolate one of the variables from each pair. In this case, t in terms of x from the first line, and s in terms of x from the second line. So we have, \(t = (x+1)/2\) and \(s = (x+12)/-5\).

Equalize the two equations

Substitute \(s\) and \(t\) into third equation for both line equations respectively, and then equate them. Giving us: \((x+1)/2 = -2((x+12)/-5) -4\).

Solve for x

Solving the equation will give us the x-coordinate of the point of intersection.

Solve for t and s

Substitute the value of x obtained from Step 4 into the respective equations of Step 2 to find the values of \(t\) and \(s\).

Solve for y and z

Substitute the value of x and t in the first line equation and the value of x and s in the second line equation to get the corresponding y and z of the point of intersection.