Solve Using an Augmented Matrix -2a+3b=15 , 4/3a+2=-2b

Problem

{[−2*a+3*b=15],[4/3*a+2=−2*b])

Solution

1. Rearrange the second equation into standard form A*a+B*b=C by adding 2*b to both sides and subtracting 2 from both sides.

4/3*a+2*b=−2

2. Clear the fraction in the second equation by multiplying every term by 3 to simplify the matrix operations.

4*a+6*b=−6

3. Write the system as an augmented matrix using the coefficients and constants from the equations.

[[−2,3,|,15],[4,6,|,−6]]

4. Eliminate the a term in the second row by replacing the second row ((R_2) with (R_2)+2*(R_1)

4+2*(−2)=0

6+2*(3)=12

−6+2*(15)=24

[[−2,3,|,15],[0,12,|,24]]

5. Normalize the second row by dividing (R_2) by 12 to solve for b

[[−2,3,|,15],[0,1,|,2]]

⇒b=2

6. Substitute the value of b back into the first row's equation, −2*a+3*b=15 to solve for a

−2*a+3*(2)=15

−2*a+6=15

−2*a=9

a=−4.5

Final Answer

(a,b)=(−4.5,2)

Want more problems? Check here!