pls with full process
Answers
(1) Suppose that L1 and L2 are lines in the plane, that the x-intercepts of L1 and L2 are 5
and −1, respectively, and that the respective y-intercepts are 5 and 1. Then L1 and L2
intersect at the point ( , ) .
(2) Consider the following system of equations.
w + x + y + z = 6
w + y + z = 4
w + 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:
x + y + z = 2
x + 3y + 3z = 0
x+ 3y+ 6z = 3
(∗)
(a) Use Gaussian elimination to put the augmented coefficient matrix into row echelon
form. The result will be
1 1 1 a
0 1 1 b
0 0 1 c
where a = , b = , and c = .
(b) Use Gauss-Jordan reduction to put the augmented coefficient matrix in reduced row
echelon form. The result will be
1 0 0 d
0 1 0 e
0 0 1 f
where d = , e = , and
f = .
(c) The solutions of (∗) are x = , y = , and z = .
(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 subset
of R
3
. Answer: .
(c) What does part (a) say about the planes x − y − 3z = 3, 2x + z = 0, and 2y + 7z