HW1_sol - CHAPTER 1 SECTION 1 1 1 1 1 1 0 2 1 2 1 0 0 4 1 2...

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CHAPTER 1 SECTION 1 2. (d) 1 1 111 021 - 21 004 1 - 2 000 1 - 3 0 0 002 5. (a) 3 x 1 +2 x 2 =8 x 1 +5 x 2 =7 (b) 5 x 1 - 2 x 2 + x 3 =3 2 x 1 +3 x 2 - 4 x 3 =0 (c) 2 x 1 + x 2 +4 x 3 = - 1 4 x 1 - 2 x 2 x 3 =4 5 x 1 x 2 +6 x 2 = - 1 (d) 4 x 1 - 3 x 2 + x 3 x 4 3 x 1 + x 2 - 5 x 3 x 4 =5 x 1 + x 2 x 3 x 4 5 x 1 + x 2 x 3 - 2 x 4 9. Given the system - m 1 x 1 + x 2 = b 1 - m 2 x 1 + x 2 = b 2 one can eliminate the variable x 2 by subtracting the frst row From the second. One then obtains the equivalent system - m 1 x 1 + x 2 = b 1 ( m 1 - m 2 ) x 1 = b 2 - b 1 1
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2 CHAPTER 1 (a) If m 1 ± = m 2 , then one can solve the second equation for x 1 x 1 = b 2 - b 1 m 1 - m 2 One can then plug this value of x 1 into the Frst equation and solve for x 2 . Thus, if m 1 ± = m 2 , there will be a unique ordered pair ( x 1 ,x 2 ) that satisFes the two equations. (b) If m 1 = m 2 , then the x 1 term drops out in the second equation 0= b 2 - b 1 This is possible if and only if b 1 = b 2 . (c) If m 1 ± = m 2 , then the two equations represent lines in the plane with different slopes. Two nonparallel lines intersect in a point. That point will be the unique solution to the system. If m 1 = m 2 and b 1 = b 2 , then both equations represent the same line and consequently every point on that line will satisfy both equations. If m 1 = m 2 and b 1 ± = b 2 , then the equations represent parallel lines. Since parallel lines do not intersect, there is no point on both lines and hence no solution to the system. 10. The system must be consistent since (0 , 0) is a solution. 11. A linear equation in 3 unknowns represents a plane in three space. The solution set to a 3 × 3 linear system would be the set of all points that lie on all three planes. If the planes are parallel or one plane is parallel to the line of intersection of the other two, then the solution set will be empty. The three equations could represent the same plane or the three planes could all intersect in a line. In either case the solution set will contain inFnitely many points. If the three planes intersect in a point then the solution set will contain only that point. SECTION 2 2. (b) The system is consistent with a unique solution (4 , - 1). 4. (b) x 1 and x 3 are lead variables and x 2 is a free variable. (d) x 1 and x 3 are lead variables and x 2 and x 4 are free variables. (f) x 2 and x 3 are lead variables and x 1 is a free variable. 5. (l) The solution is (0 , - 1 . 5 , - 3 . 5). 6. (c) The solution set consists of all ordered triples of the form (0 , - α,α ). 7. A homogeneous linear equation in 3 unknowns corresponds to a plane that passes through the origin in 3-space. Two such equations would correspond to two planes through the origin. If one equation is a multiple of the other, then both represent the same plane through the origin and every point on that plane will be a solution to the system. If one equation is not a multiple of the other, then we have two distinct planes that intersect in a line through the origin. Every point on the line of intersection will be a solution to the linear system. So in either case the system must have inFnitely many solutions.
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HW1_sol - CHAPTER 1 SECTION 1 1 1 1 1 1 0 2 1 2 1 0 0 4 1 2...

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