76 CHAPTER 4. SYSTEMS OF EQUATIONS
25. Find the general solution of the system whose augmented matrix is1 0 2 1 1 | 20 1 0 1 2 | 10 2 0 0 1 | 31 −1 2 2 2 | 0
.
26. Give the complete solution to the system of equations, 7x+ 14y+ 15z = 22, 2x+4y+3z = 5, and 3x+6y+10z = 13.
27. Give the complete solution to the system of equations, 3x− y+ 4z = 6, y+ 8z = 0,and −2x+ y =−4.
28. Give the complete solution to the system of equations, 9x− 2y+ 4z = −17, 13x−3y+6z =−25, and −2x− z = 3.
29. Give the complete solution to the system of equations, 65x+84y+16z = 546, 81x+105y+20z = 682, and 84x+110y+21z = 713.
30. Give the complete solution to the system of equations, 8x+2y+3z =−3,8x+3y+3z =−1, and 4x+ y+3z =−9.
31. Give the complete solution to the system of equations, −8x+ 2y+ 5z = 18,−8x+3y+5z = 13, and −4x+ y+5z = 19.
32. Give the complete solution to the system of equations, 3x− y− 2z = 3, y− 4z = 0,and −2x+ y =−2.
33. Give the complete solution to the system of equations,−9x+15y= 66,−11x+18y=79 ,−x+ y = 4, and z = 3.
34. Give the complete solution to the system of equations, −19x+8y = −108, −71x+30y =−404, −2x+ y =−12, 4x+ z = 14.
35. Consider the system −5x+ 2y− z = 0 and −5x− 2y− z = 0. Both equations equalzero and so −5x+ 2y− z = −5x− 2y− z which is equivalent to y = 0. Thus x andz can equal anything. But when x = 1, z = −4, and y = 0 are plugged in to theequations, it doesn’t work. Why?
36. Four times the weight of Gaston is 150 pounds more than the weight of Ichabod.Four times the weight of Ichabod is 660 pounds less than seventeen times the weightof Gaston. Four times the weight of Gaston plus the weight of Siegfried equals 290pounds. Brunhilde would balance all three of the others. Find the weights of the foursisters.
37. The steady state temperature, u in a plate solves Laplace’s equation, ∆u= 0. One wayto approximate the solution which is often used is to divide the plate into a squaremesh and require the temperature at each node to equal the average of the temperatureat the four adjacent nodes. This procedure is justified by the mean value property ofharmonic functions. In the following picture, the numbers represent the observedtemperature at the indicated nodes. Your task is to find the temperature at the interiornodes, indicated by x,y,z, and w. One of the equations is z = 1
4 (10+0+w+ x).