Solve Using a Matrix by Elimination 5a+3b+d=90 , 2a+4b+g=64

Problem

{[5*a+3*b+d=90],[2*a+4*b+g=64])

Solution

1. Write the system as an augmented matrix representing the coefficients of a b d and g

[[5,3,1,0,|,90],[2,4,0,1,|,64]]

2. Normalize the first row by dividing all elements in (R_1) by 5 to create a leading 1

[[1,0.6,0.2,0,|,18],[2,4,0,1,|,64]]

3. Eliminate the a term from the second row by performing the row operation (R_2)−2*(R_1)→(R_2)

[[1,0.6,0.2,0,|,18],[0,2.8,−0.4,1,|,28]]

4. Normalize the second row by dividing (R_2) by 2.8 to create a leading 1 for the b column.

[[1,0.6,0.2,0,|,18],[0,1,−1/7,5/14,|,10]]

5. Eliminate the b term from the first row by performing the row operation (R_1)−0.6*(R_2)→(R_1)

[[1,0,2/7,−3/14,|,12],[0,1,−1/7,5/14,|,10]]

6. Identify the pivot variables a and b and the free variables d and g to express the general solution.

a=12−2/7*d+3/14*g

b=10+1/7*d−5/14*g

Final Answer

{[a=12−2/7*d+3/14*g],[b=10+1/7*d−5/14*g])

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