INSTRUCTIONS:

1) Do all necessary steps

2) This is a whole paper. No data is missing.

1.2. Exercises (1) Suppose that In and Ly are lines in the plane, that the r-intercepts of Ly and La are 5 and -1, respectively, and that the respective y-intercepts are 5 and 1. Then L, and Ly intersect at the point ( _. - ) . (2) Consider the following system of equations. +#+= =4 (# ) + y =2 (a) List the leading variables (b) List the free variables (c) The general solution of (*) (expressed in terms of the free variables) is (d) Suppose that a fourth equation -2w + y = 5 is included in the system (*). What is the solution of the resulting system? Answer: ( _ _ (e) Suppose that instead of the equation in part (d), the equation -2w - 2y = -3 is included in the system (*). Then what can you say about the solution(s) of the resulting system? Answer: (3) Consider the following system of equations; 3+ 7+ 2=2 + + 37 + 32 = 0 (* ) at 3y+ 62 = 3 (a) Use Gaussian elimination to put the augmented coefficient matrix into row echelon 1 1 form. The result will be 0 1 1 b where a = _ , and c = 101 C (b) Use Gauss-Jordan reduction to put the augmented coefficient matrix in reduced row 100d echelon form. The result will be e where d = - , and 0 01 5 f = _ (c) The solutions of (*) are = = -, and = = (4) Consider the following system of equations, 0.003000x + 59.14y = 59.17 5.291x - 6.130y = 46.78. (a) Using only row operation III and back substitution find the exact solution of the system. Answer: : = (b) Same as (a), but after performing each arithmetic operation round off your answer to four significant figures, Answer: I = 1 =(5) Find the values of & for which the system of equations (atky = 1 krty = 1 has (a) no solution. Answer: (b) exactly one solution. Answer: (c) infinitely many solutions. Answer: (d) When there is exactly one solution, it is r = and y = (6) Consider the following two systems of equations. atytz=6 x + 2y + 2= = 11 (1) 2x + 3y - 42 = 3 and at y+ z= 7 + + 20+ 28 = 10 (2) 2r + 3y - 4z = 3 Solve both systems simultaneously by applying Gauss-Jordan reduction to an appro- priate 3 x 5 matrix. (a) The resulting row echelon form of this 3 x 5 matrix is (b) The resulting reduced row echelon form is (c) The solution for (1) is (_ ) and the solution for (2) is (. (7) Consider the following system of equations: C - y- 32 = 3 + 2=0 20 + 7: = c (a) For what values of c does the system have a solution? Answer: c= (b) For the value of c you found in (a) describe the solution set geometrically as a subset of R"