Solve if your are 100 % sure. 4. Find the following linear equations. We want to

Question

Solve if your are 100 % sure.

4. Find the following linear equations. We want to solve them by substitution -a +5b-7c+d--21) a+4d -5...(2) -b-d3) -c-2d34) 4.1. Solve Eq. (1) for a in terms of b, c, d; substitute a in Eq. (2) and solve Eq. (2) for b in terms of c and d; substitute b in Eq. (3) and solve Eq. (3) for c in terms of d; and substitute c in Eq. (4) and solve Eq. (4) for d. After obtaining d evaluate a, b, and c. 4.2. Solve Eqs. (2), (3), and (4) for a, b, c in terms of d. Solve (1) for d by substitution.

Explanation / Answer

4. We have –a+5b -7c+d = -2…(1)

                      a+4d = 5...(2)

                     -b-d = -2 or, b+d = 2…(3)

                     -c-2d = -3 or, c+2d = 3…4)

From the 4th equation, we get c = 3-2d. From the 3rd equation, we get b = 2-d, and from the 3nd equation, we get a = 5-4d. On substituting these values of a,b,c in the 1st equation, we get –(5-4d)+5(2-d)-7(3-2d)+d =2 or,-5+4d +10-5d-21+14d+d=2 or, -16+14d=2 or,14d=2+16=18 so that d = 18/14 = 9/7. Then a = 5-4*(9/7) = 5-36/7= -1/7, b = 2-9/7 = 5/7, and c = 3-2*(9/7) = 3-18/7 = 3/7. Thus, a = -1/7, b = 5/7, c = 3/7 and d = 9/7.

4.1 From the 1st equation, we get a = 5b -7c+d -2. On substituting this value of a in the 2nd equation, we get 5b -7c+d -2 +4d = 5 or, 5b -7c+5d= 7 or, 5b = 7+7c-5d or, b = 7/5 +7c/5 –d. On substituting this value of b in the 3rd equation, we get 7/5 +7c/5 –d +d = 2 or, 7c/5 = 2-7/5 = 3/5 so that c = (3/5)*(5/7) = 3/7. On substituting this value of c in the 4th   equation, we get 3/7 +2d = 3 or, 2d = 3-3/7 = 18/7 so that d = 9/7. Then b = 7/5 +7c/5 –d = 7/5 +(7/5)(3/7) -9/7 = 7/5 +3/5-9/7 = 5/7 and a = 5b -7c+d -2 = 5*(5/7)-7(3/7)+9/7-2 = 25/7 +9/7-5 = -1/7.

4.2. Already done in 4 above.

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