WK8HW - JO - WK8HW - JO Section 6.2 5. x + 2y + z = 5 2x +...

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Unformatted text preview: WK8HW - JO Section 6.2 5. x + 2y + z = 5 2x + y - 3z = -2 3x + y + 4z = -5 equation 1 equation 2 equation 3 x 1 2 3 x 1 0 3 x 1 0 0 x 1 0 0 x 1 0 0 x 1 0 0 x 1 0 0 x 1 0 0 x 1 0 0 y 2 1 1 y 2 -3 1 y 2 -3 -5 y 2 1 -5 y 2 1 0 y 2 1 0 y 0 1 0 y 0 1 0 y 0 1 0 z 1 -3 4 z 1 -5 4 z 1 -5 1 z 1 5/3 1 z 1 5/3 28/3 z 1 5/3 28/3 z 7/3 5/3 28/3 z 7/3 5/3 1 z 0 5/3 1 constant 5 -2 -5 constant 5 -12 -5 constant 5 -12 -20 constant 5 4 -20 constant 5 4 0 constant 5 4 0 constant -3 4 0 constant -3 4 0 constant -3 4 0 equation 1 equation 2 equation 3 - 2 row 1 + row 2 to make 2x equal 0 equation 1 equation 2 equation 3 - 3 row 1 + row 3 to make 3x equal 0 equation 1 equation 2 equation 3 - 1/3 row 2 to make -3 y equal 1 equation 1 equation 2 equation 3 5 row 2 + row 3 to make -5y equal to 0 equation 1 equation 2 equation 3 - 2 row 2 + row 1 to make 2y equal to 0 equation 1 equation 2 equation 3 3/28 row 3 to make 28/3z equal 1 equation 1 equation 2 equation 3 - 7/3 row 3 + row 1 to make 7/3 equal to 0 equation 1 equation 2 equation 3 - 5/3 row 3 + row 2 to make 5/3 equal to 0 equation 1 equation 2 equation 3 x 1 0 0 y 0 1 0 z 0 0 1 constant -3 4 0 the solution is (-3, 4, 0) 11. 3x + 5y -z = 0 4x - y + 2z = 1 6x - 10y + 2z = 0 x 3 4 -6 x 3 4 0 x 1 4 0 x 1 0 0 x 1 0 0 x 1 0 0 y 5 -1 -10 y 5 -1 0 y 5/3 -1 0 y 5/3 23/3 0 y 5/3 1 0 y 0 1 0 z -1 2 2 z -1 2 0 z 1/3 2 0 z 1/3 10/3 0 z 1/3 10/3 0 z 9/23 10/3 0 constant 0 1 0 constant 0 1 0 constant 0 1 0 constant 0 1 0 constant 0 3/23 0 constant 5/23 3/23 0 equation 1 equation 2 equation 3 2 row 1 + row 3 to make -6x equal to 0 equation 1 equation 2 equation 3 1/3 row 1 to make 3x equal to 1 equation 1 equation 2 equation 3 - 4 row 1 + row 2 to make 4x equal to 0 equation 1 equation 2 equation 3 - 3/23 row 2 to make 23/3y equal to 1 equation 1 equation 2 equation 3 - 5/3 row 2 + row 1 to make 5/3y equal to 0 equation 1 equation 2 equation 3 1x + 9/23z = 5/23 x = 5/23 - 9z/23 x= 5 - 9z 23 y= 19. -8x - 9y = 11 24x + 34y = 2 16x + 11y = -57 x -8 24 16 y -9 34 11 constant 11 2 -57 equation 1 equation 2 equation 3 equation 1 equation 2 equation 3 x -8 0 16 x -8 0 0 x -8 0 0 x -8 0 0 x 1 0 0 x 1 0 0 x 1 0 0 y -9 7 11 y -9 7 -7 y -9 7 -7 y -9 1 1 y 9/8 1 1 y 0 1 1 y 0 1 0 constant 11 35 -57 constant 11 35 -35 constant 11 35 -35 constant 11 5 5 constant 11/8 5 5 constant -7 5 5 constant -7 5 0 3 row 1 + row 2 new row 2 to make 24x equal to 0 equation 1 equation 2 equation 3 new row 3 2 row 1 + row 3 to make 16x equal to 0 new row 2 1/7 row 2 new row 3 - 1/7 row 3 to make 7y equal to 1 and to make -7 equal to 1 - 1/8 row 1 new row 1 to make -8 equal to 1 equation 1 equation 2 equation 3 equation 1 equation 2 equation 3 equation 1 equation 2 equation 3 - 9/8 row 2 + row 1 to make 9/8y equal to 0 new row 1 equation 1 equation 2 equation 3 - 1 row 2 + row 3 new row 3 to make 1y equal to 0 equation 1 equation 2 equation 3 the solution is (-7, 5) 25. x + y + z + w = -5 4x + 3y + z - w = -2 2x + y + 3z - 2w = -6 2x - 2y + 2z + 2w = -10 x 1 4 2 -2 x 1 4 I interchanged rows 1 and 3 so that equation 1 would have a 1 in the first box equation 1 equation 2 equation 3 equation 4 y 1 3 1 -2 y 1 3 z 1 1 3 2 z 1 1 w 1 -1 -2 2 w 1 -1 constant -5 -2 -6 -10 constant -5 -2 equation 1 equation 2 - 4 row 1 + row 2 equation 3 equation 4 2 -2 x 1 0 2 -2 x 1 0 0 -2 x 1 0 0 0 x 1 0 0 0 x 1 0 0 0 x 1 0 0 0 x 1 0 0 0 x 1 0 0 0 1 -2 y 1 -1 1 -2 y 1 -1 -1 -2 y 1 -1 -1 0 y 1 1 -1 0 y 0 1 0 0 y 0 1 0 0 y 0 1 0 0 y 0 1 0 0 3 2 z 1 -3 3 2 z 1 -3 1 2 z 1 -3 1 4 z 1 3 1 4 z -2 3 4 4 z -2 3 1 4 z 0 3 1 4 z 0 0 1 4 -2 2 w 1 -5 -2 2 w 1 -5 -4 2 w 1 -5 -4 4 w 1 5 -4 4 w -4 5 1 4 w -4 5 1/4 4 w 7/2 5 1/4 4 w 7/2 17/4 1/4 4 -6 -10 constant -5 18 -6 -10 constant -5 18 4 -10 constant -5 18 4 -20 constant -5 -18 4 -20 constant 13 -18 -14 -20 constant 13 -18 7/2 -20 constant 6 -18 7/2 -20 constant 6 15/2 7/2 -20 to make 4x equal to 0 equation 1 equation 2 equation 3 equation 4 - 2 row 1 + row 3 to make 2x equal to 0 equation 1 equation 2 equation 3 equation 4 2 row 1 + row 4 to make -2x equal to 0 equation 1 equation 2 equation 3 equation 4 - 1 row 2 to make -1y equal to 1 equation 1 equation 2 equation 3 equation 4 - 1 row 2 + row 1 to make 1y equal to 0 equation 1 equation 2 equation 3 equation 4 1 row 2 + row 3 to make -1y equal to 0 equation 1 equation 2 equation 3 equation 4 1/4 row 3 to make 4z equal to 1 equation 1 equation 2 equation 3 equation 4 2 row 3 + row 1 to make -2z equal 0 equation 1 equation 2 equation 3 equation 4 - 3 row 3 + row 2 to make 3z equal to 0 equation 1 equation 2 equation 3 equation 4 x 1 0 0 0 x 1 0 0 0 x 1 0 0 0 x 1 0 0 0 x 1 0 0 0 x 1 0 0 0 y 0 1 0 0 y 0 1 0 0 y 0 1 0 0 y 0 1 0 0 y 0 1 0 0 y 0 1 0 0 z 0 0 1 0 z 0 0 1 0 z 0 0 1 0 z 0 0 1 0 z 0 0 1 0 z 0 0 1 0 w 7/2 17/4 1/4 3 w 7/2 17/4 1/4 1 w 0 17/4 1/4 1 w 0 0 1/4 1 w 0 0 0 1 w 0 0 0 1 constant 6 15/2 7/2 -6 constant 6 15/2 7/2 -2 constant -1 15/2 7/2 -2 constant -1 1 7/2 -2 constant -1 1 -3 -2 constant -1 1 -3 -2 - 4 row 3 + row 4 to make 4z equal to 0 equation 1 equation 2 equation 3 equation 4 1/3 row 4 to make 3w equal 1 7/2 row 4 + row 1 to make -7/2w equal 0 equation 1 equation 2 equation 3 equation 4 equation 1 equation 2 equation 3 equation 4 - 17/4 row 4 + row 2 to make 17/4w equal 0 equation 1 equation 2 equation 3 equation 4 - 1/4 row 4 + row 3 to make 1/4w equal to 0 equation 1 equation 2 equation 3 equation 4 the solution is (-1, 1, -3, -2) 31. x = number of hours for the Garcia firm y = number of hours fror the Wong firm Garcia firm 10 phone surveys 30 mail surveys 5 interviews/hour Wong firm 20 phone surveys 10 mail surveys 10 interviews/hour need 500 phone surveys need 750 mail surveys need 250 in person interviews 10x + 20y = 500 30x + 10y = 750 5x + 10y = 250 x 10 30 5 x 1 0 5 x 1 0 0 x 1 0 0 x 1 0 0 x 1 0 0 y 20 10 10 y 2 -50 10 y 2 -50 0 y 2 1 0 y 0 1 0 y 0 1 0 constant 500 750 250 constant 50 -750 250 constant 50 -750 0 constant 50 -15 0 constant 20 15 0 constant 20 15 0 new row 1 1/10 row 1 equation 1 equation 2 equation 3 equation 1 equation 2 equation 3 - 30 row 1 + row 2 new row 2 equation 1 equation 2 equation 3 new row 3 - 5 row 1 + row 3 equation 1 equation 2 equation 3 - 1/50 row 2 new row 2 equation 1 equation 2 equation 3 - 2 row 2 + row 1 new row 1 equation 1 equation 2 equation 3 the solution is (20, 15) So, the Garcia firm should be hired to for 20 hours and new row 2 -2 2 0 -4 1 -3 -2 -3 -5 -10 -2 -12 new row 3 -3 3 0 -6 1 -5 -3 4 1 -15 -5 -20 new row 2 0 0 0 5 -5 0 25/3 1 28/3 20 -20 0 new row 3 new row 1 0 1 1 -2 2 0 10/3 1 7/3 -8 5 -3 red numbers means the fractions are negative new row 3 0 1 new row 1 1 0 0 0 7/3 7/3 0 0 -3 -3 new row 2 0 0 0 0 1 1 5/3 5/3 0 0 4 4 new row 3 6 -6 0 10 -10 0 -2 2 0 0 0 0 new row 1 3/3 1 5/3 5/3 1/3 1/3 0 0 new row 2 -4 4 0 20/3 -1 23/3 4/3 2 10/3 0 1 1 new row 2 equal to 1 0 1 10/3 3/23 new row 1 0 1 1 5/3 5/3 0 50/69 1/3 9/23 5/23 0 5/23 1y - 10/3z = -3/23 y = -3/23 + 10z/23 10z - 3 23 -24 24 0 -27 34 7 33 2 35 -16 16 0 -18 11 -7 22 -57 -35 0 0 to make -7 equal to 1 1 1 1 5 5 9/8 11/8 0 1 1 9/8 9/8 0 45/8 11/8 56/8 56/8 = -7 0 0 0 -1 1 0 -5 5 0 w 1 + row 2 new row 2 -4 4 0 -4 3 -1 -4 1 -3 -4 -1 -5 20 -2 18 ake 4x equal to 0 w 1 + row 3 ake 2x equal to 0 new row 3 -2 2 0 -2 1 -1 -2 3 1 -2 -2 -4 10 -6 4 new row 4 w 1 + row 4 ake -2x equal to 0 2 -2 0 2 -2 0 2 2 4 2 2 4 -10 -10 -20 new row 2 ake -1y equal to 1 0 1 3 5 -18 w 2 + row 1 ake 1y equal to 0 new row 1 0 1 1 -1 1 0 -3 1 -2 -5 1 -4 18 -5 13 new row 3 w 2 + row 3 ake -1y equal to 0 0 0 0 1 -1 0 3 1 4 5 -4 1 -18 4 -14 new row 3 ake 4z equal to 1 0 0 4/4 1/4 7/2 w 3 + row 1 ake -2z equal 0 new row 1 0 1 1 0 0 0 2 -2 0 1/2 -4 7/2 -7 13 6 w 3 + row 2 ake 3z equal to 0 new row 2 0 0 0 0 1 1 -3 3 0 3/4 5 17/4 21/2 -18 15/2 new row 4 w 3 + row 4 ake 4z equal to 0 0 0 0 0 0 0 -4 4 0 -1 4 3 14 -20 -6 new row 4 ake 3w equal 1 row 4 + row 1 ake -7/2w equal 0 new row 1 0 0 0 1 -2 0 1 1 0 0 0 0 0 0 7/2 7/2 0 7 6 -1 4 row 4 + row 2 new row 2 ake 17/4w equal 0 0 0 0 0 1 1 0 0 0 17/4 17/4 0 17/2 15/2 1 new row 3 row 4 + row 3 ake 1/4w equal to 0 0 0 0 0 0 0 0 1 1 1/4 1/4 0 1/2 7/2 -3 solution is (-1, 1, -3, -2) 1 2 50 -30 30 0 -60 10 -50 -1500 750 -750 -5 5 0 -10 10 0 -250 250 0 0 1 15 0 1 1 -2 2 0 -30 50 20 be hired to for 20 hours and the Wong firm should be hired for 15 hours ...
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