Solve Using a Matrix by Elimination -5a-30b+5d=-10 , a+4b-5d=6 , -a-4b+9d=-3

Problem

{[−5*a−30*b+5*d=−10],[a+4*b−5*d=6],[−a−4*b+9*d=−3])

Solution

1. Write the system of equations as an augmented matrix.

[[−5,−30,5,|,−10],[1,4,−5,|,6],[−1,−4,9,|,−3]]

2. Simplify the first row by dividing by −5 to create a leading 1

[[1,6,−1,|,2],[1,4,−5,|,6],[−1,−4,9,|,−3]]

3. Eliminate the entries below the first leading 1 by performing (R_2)−(R_1)→(R_2) and (R_3)+(R_1)→(R_3)

[[1,6,−1,|,2],[0,−2,−4,|,4],[0,2,8,|,−1]]

4. Simplify the second row by dividing by −2 to create the next leading 1

[[1,6,−1,|,2],[0,1,2,|,−2],[0,2,8,|,−1]]

5. Eliminate the entry below the second leading 1 by performing (R_3)−2*(R_2)→(R_3)

[[1,6,−1,|,2],[0,1,2,|,−2],[0,0,4,|,3]]

6. Solve for d using the third row.

4*d=3

d=3/4

7. Substitute d into the second row equation to solve for b

b+2*(3/4)=−2

b+3/2=−2

b=−7/2

8. Substitute b and d into the first row equation to solve for a

a+6*(−7/2)−3/4=2

a−21−3/4=2

a=23+3/4

a=95/4

Final Answer

(a,b,d)=(95/4,−7/2,3/4)

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