Question

Let \(a,b\in\mathbb{R}\). If the system of linear equations

Hint:

For infinitely many solutions in a linear system, one equation must become a linear combination of the others when the first two equations are independent.

Answer 1

Correct Answer: 6

Step 1: When are there infinitely many solutions.
For an infinite family of solutions, the third equation must be a mix of the first two.

Step 2: Set up the mix.
Suppose the third row equals $\lambda$ times the first plus $\mu$ times the second, including the constant on the right.

Step 3: Match $y$ and constants.
Matching the $y$ coefficient gives $5=2\lambda+3\mu$. Matching the constant gives $b=2\lambda+\mu$ and also $b=\lambda+2\mu$.

Step 4: Solve for $\lambda,\mu$.
From the two forms of $b$, $\lambda=\mu$. Then $5=5\lambda$ gives $\lambda=\mu=1$.

Step 5: Find $a$ and $b$.
So $a=\lambda+2\mu=3$ and $b=\lambda+2\mu=3$.

Step 6: Add.
\[ a+b=3+3=6 \]
\[ \boxed{6.0} \]