Solve Using a Matrix by Elimination

Problem

{[4*a+3*b+c=4],[25*a+5*b+c=4],[4*a+2*b+c=1])

Solution

1. Identify the system of linear equations by evaluating the squared terms. The system is:

4*a+3*b+c=4

25*a+5*b+c=4

4*a+2*b+c=1

2. Write the augmented matrix representing the system:

[[4,3,1,|,4],[25,5,1,|,4],[4,2,1,|,1]]

3. Eliminate the a term in the third row by subtracting the first row from the third row ((R_3)−(R_1)→(R_3):

[[4,3,1,|,4],[25,5,1,|,4],[0,−1,0,|,−3]]

4. Solve for b using the third row:

−b=−3

b=3

5. Eliminate the c term by subtracting the first row from the second row ((R_2)−(R_1)→(R_2):

21*a+2*b=0

6. Substitute b=3 into the equation 21*a+2*b=0 to solve for a

21*a+2*(3)=0

21*a=−6

a=−6/21

a=−2/7

7. Substitute a=−2/7 and b=3 into the first equation to solve for c

4*(−2/7)+3*(3)+c=4

−8/7+9+c=4

c=4−9+8/7

c=−5+8/7

c=−35/7+8/7

c=−27/7

Final Answer

(a,b,c)=(−2/7,3,−27/7)

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