Chapter Review Exercises443We want to find the value oftfor which the point(x,y,z)lies on the plane. We substitute the parametric equations inthe equation of the plane and solve fort:2(3t+2)−3·1+(−7t)=56t+4−3−7t=5−t=4⇒t= −4The pointPof intersection of the line and the plane has the coordinates:x=3·(−4)+2= −10,y=1,z= −7·(−4)=28and thus,P=(−10,1,28) .47.Find the trace of the plane 3x−2y+5z=4 in thexy-plane.SOLUTIONThexy-plane has equationz=0, therefore the intersection of the plane 3x−2y+5z=4 with thexy-plane must satisfy bothz=0 and the equation of the plane. Therefore the trace has the following equation:3x−2y+5·0=4⇒3x−2y=4We conclude that the trace of the plane in thexy-plane is the line 3x−2y=4 in thexy-plane.48.Find the intersection of the planesx+y+z=1 and 3x−2y+z=5.SOLUTIONThe line of intersection of the planesx+y+z=1 and 3x−2y+z=5 consists of all points that satisfythe equations of the two planes. Therefore, we must solve the following equations:x+y+z=13x−2y+z=5The first equation impliesy=1−x−z. Substituting in the second equation and solving forxin terms ofz
