Solve Using a Matrix by Elimination a+2b+4d=13 , -a+b+2d=2 , 4a+10b+12d=46

Problem

{[a+2*b+4*d=13],[−a+b+2*d=2],[4*a+10*b+12*d=46])

Solution

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

[[1,2,4,|,13],[−1,1,2,|,2],[4,10,12,|,46]]

2. Eliminate the a terms from the second and third rows using the first row. Add (R_1) to (R_2) ((R_2)+(R_1)⇒(R_2) and subtract 4 times (R_1) from (R_3) ((R_3)−4*(R_1)⇒(R_3).

[[1,2,4,|,13],[0,3,6,|,15],[0,2,−4,|,−6]]

3. Simplify the second row by dividing by 3 ((R_2)/3⇒(R_2).

[[1,2,4,|,13],[0,1,2,|,5],[0,2,−4,|,−6]]

4. Eliminate the b term from the third row using the second row ((R_3)−2*(R_2)⇒(R_3).

[[1,2,4,|,13],[0,1,2,|,5],[0,0,−8,|,−16]]

5. Solve for d using the third row.

−8*d=−16

d=2

6. Substitute d=2 into the second row equation to solve for b

b+2*(2)=5

b+4=5

b=1

7. Substitute b=1 and d=2 into the first row equation to solve for a

a+2*(1)+4*(2)=13

a+2+8=13

a+10=13

a=3

Final Answer

(a,b,d)=(3,1,2)

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