Solve Using a Matrix by Elimination 0.15g-0.35h=-0.5 , -0.012g+0.25h=0.1

Problem

{[0.15*g−0.35*h=−0.5],[−0.012*g+0.25*h=0.1])

Solution

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

[[0.15,−0.35,|,−0.5],[−0.012,0.25,|,0.1]]

2. Multiply the first row (R_1) by 1/0.15 to create a leading 1 in the first column.

[[1,−7/3,|,−10/3],[−0.012,0.25,|,0.1]]

3. Eliminate the g term in the second row by performing the row operation (R_2)+0.012*(R_1)⇒(R_2)

[[1,−7/3,|,−10/3],[0,0.222,|,0.06]]

4. Simplify the second row by converting decimals to fractions to find 0.222=2/9 and 0.06=3/50

[[1,−7/3,|,−10/3],[0,2/9,|,3/50]]

5. Multiply the second row (R_2) by 9/2 to solve for h

[[1,−7/3,|,−10/3],[0,1,|,27/100]]

6. Substitute the value of h=0.27 back into the first row equation g−7/3*h=−10/3 to solve for g

g−7/3*(0.27)=−10/3

g−0.63=−3.333...

g=−2.70333...

7. Calculate the exact fractional values for the variables.

h=27/100

g=−10/3+7/3*(27/100)

g=−10/3+63/100

g=−1000/300+189/300

g=−811/300

Final Answer

(g,h)=(−811/300,27/100)

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