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To solve bài 2b, we need to find the value of \( m \) such that the system of equations has exactly one solution \((x, y)\) that satisfies \( x^2 + y^2 = 5 \). The system of equations is:

\[
\begin{cases}
3x - y = 2m + 3 \\
x + 2y = 3m + 1
\end{cases}
\]

First, solve this system of linear equations:

1. Multiply the first equation by 2 to align with the \( y \) terms in the second equation:
\[
\begin{cases}
6x - 2y = 4m + 6 \\
x + 2y = 3m + 1
\end{cases}
\]

2. Add the two equations to eliminate \( y \):
\[
6x - 2y + x + 2y = 4m + 6 + 3m + 1 \\
7x = 7m + 7 \\
x = m + 1
\]

3. Substitute \( x = m + 1 \) back into one of the original equations to find \( y \). Using \( x + 2y = 3m + 1 \):
\[
(m + 1) + 2y = 3m + 1 \\
2y = 2m \\
y = m
\]

So, the solution to the system is \( x = m + 1 \) and \( y = m \).

Next, we need \( x^2 + y^2 = 5 \):
\[
(m + 1)^2 + m^2 = 5 \\
m^2 + 2m + 1 + m^2 = 5 \\
2m^2 + 2m + 1 = 5 \\
2m^2 + 2m - 4 = 0 \\
m^2 + m - 2 = 0
\]

Solve this quadratic equation:
\[
m^2 + m - 2 = 0 \\
(m + 2)(m - 1) = 0
\]

So, \( m = -2 \) or \( m = 1 \).

Thus, the values of \( m \) that satisfy the condition are \( m = -2 \) and \( m = 1 \).

Would you like a detailed explanation for another question or have any further questions?

Here are 5 additional questions you might be interested in:

1. How to solve the system of equations in bài 1a?
2. What is the solution to the quadratic equation in bài 3a when \( m = 4 \)?
3. How do you determine if a system of linear equations has a unique solution?
4. What are the steps to solve bài 1c?
5. Can you explain how to solve bài 1d with fractions?

**Tip:** When solving systems of equations, always check the determinant (if applicable) to ensure there is a unique solution before proceeding with substitution or elimination methods.

Learn how to solve a system of linear equations with a quadratic condition. Step-by-step solution to find the values of m for a unique solution that satisfies x^2 + y^2 = 5. Suitable for high school students studying algebra and systems of equations.

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