https://file.aiquickdraw.com/mathbot/file/17204931850167u9nj8n4.jpg

To solve part \( b \) of Bài 2, we need to find the value of \( m \) such that the system of equations:

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

has exactly one solution \((x, y)\) that satisfies \( x^2 + y^2 = 5 \).

### Step 1: Find the conditions for a unique solution
A system of linear equations has a unique solution if the determinant of the coefficient matrix is non-zero. The coefficient matrix for this system is:

\[ \begin{pmatrix}
3 & -1 \\
1 & 2
\end{pmatrix} \]

The determinant \(\Delta\) is calculated as:

\[ \Delta = 3 \cdot 2 - (-1) \cdot 1 = 6 + 1 = 7 \]

Since \(\Delta = 7 \neq 0\), the system always has a unique solution for any value of \( m \).

### Step 2: Solve the system for general \( m \)
Using Cramer's Rule, we find the solution \( (x, y) \).

First, let's find the determinants of the matrices formed by replacing the respective columns with the constants:

\[ \Delta_x = \begin{vmatrix}
2m + 3 & -1 \\
3m + 1 & 2
\end{vmatrix} = (2m + 3) \cdot 2 - (-1) \cdot (3m + 1) = 4m + 6 + 3m + 1 = 7m + 7 \]

\[ \Delta_y = \begin{vmatrix}
3 & 2m + 3 \\
1 & 3m + 1
\end{vmatrix} = 3 \cdot (3m + 1) - 1 \cdot (2m + 3) = 9m + 3 - 2m - 3 = 7m \]

So the solutions are:

\[ x = \frac{\Delta_x}{\Delta} = \frac{7m + 7}{7} = m + 1 \]

\[ y = \frac{\Delta_y}{\Delta} = \frac{7m}{7} = m \]

### Step 3: Check the condition \( x^2 + y^2 = 5 \)
Substituting \( x = m + 1 \) and \( y = m \):

\[ (m + 1)^2 + m^2 = 5 \]
\[ m^2 + 2m + 1 + m^2 = 5 \]
\[ 2m^2 + 2m + 1 = 5 \]
\[ 2m^2 + 2m - 4 = 0 \]
\[ m^2 + m - 2 = 0 \]

Solving this quadratic equation:

\[ m^2 + m - 2 = (m + 2)(m - 1) = 0 \]

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

Thus, the values of \( m \) that ensure the system has a unique solution \((x, y)\) satisfying \( x^2 + y^2 = 5 \) are \( m = -2 \) or \( m = 1 \).

Do you have any questions or need further details? Here are 5 more questions you might want to ask:

1. Can you solve part (a) of Bài 2 for me?
2. How do you solve part (a) of Bài 3?
3. Can you help with solving the systems of equations in Bài 1?
4. What method can be used to solve non-linear systems of equations?
5. How do you determine if a system of equations has no solution, one solution, or infinitely many solutions?

**Tip:** When solving systems of equations, always check the determinant first to ensure the system has a unique solution.

Learn how to solve a system of linear equations to find a unique solution that satisfies specific conditions. This detailed guide covers the application of Cramer's Rule and determinant calculations for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation