Find the Variables

Problem

[[−2,−5],[−2,−1]]*[[a],[b]]+[[−19],[11]]=[[−43],[−5]]

Solution

1. Perform matrix multiplication on the left side to combine the first two terms.

[[−2*a−5*b],[−2*a−b]]+[[−19],[11]]=[[−43],[−5]]

2. Add the vectors on the left side by summing their corresponding components.

[[−2*a−5*b−19],[−2*a−b+11]]=[[−43],[−5]]

3. Create a system of linear equations by setting the corresponding entries equal to each other.

−2*a−5*b−19=−43

−2*a−b+11=−5

4. Simplify the equations by moving the constant terms to the right side.

−2*a−5*b=−24

−2*a−b=−16

5. Subtract the second equation from the first equation to eliminate the variable a

(−2*a−5*b)−(−2*a−b)=−24−(−16)

−4*b=−8

6. Solve for b by dividing both sides by −4

b=2

7. Substitute the value of b back into the simplified second equation to find a

−2*a−2=−16

−2*a=−14

a=7

Final Answer

a=7,b=2

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