https://file.aiquickdraw.com/mathbot/file/1726680801357tpw6fyp9.jpg

The system of linear equations given in the image is:

\[
\begin{cases} 
3x + 2y = 1 \\
bx + 4y = k
\end{cases}
\]

Here, \( b \) and \( k \) are constants. To solve this system of equations, we can use methods like substitution, elimination, or matrix methods (e.g., Cramer's rule). However, without specific values for \( b \) and \( k \), we can only provide a general solution in terms of \( b \) and \( k \).

### 1. Solving the System Using Substitution or Elimination
Let's proceed with the elimination method to eliminate one of the variables.

#### Step 1: Eliminate \( x \)
Multiply the first equation by \( b \) to align the coefficients of \( x \):

\[
\begin{cases} 
b(3x + 2y) = b \times 1 \\
bx + 4y = k
\end{cases}
\]

This simplifies to:

\[
\begin{cases} 
3bx + 2by = b \\
bx + 4y = k
\end{cases}
\]

Now subtract the second equation from the first:

\[
(3bx - bx) + (2by - 4y) = b - k
\]

\[
2bx - 4y(b-2) = b - k
\]

#### Step 2: Solve for One Variable
This leaves us with one equation in terms of \( x \) and \( y \). To express the solution, more specific values or conditions for \( b \) and \( k \) are needed.

### Special Cases:
1. **Unique Solution:** If \( b \neq 3 \), the system has a unique solution.
2. **Infinite Solutions:** If the two lines are coincident (exact same line), this can be determined by comparing slopes and intercepts.
3. **No Solution:** If the lines are parallel but not coincident, the system has no solution.

Do you want to solve this system for a specific case (specific values of \( b \) and \( k \)), or are you looking for a general approach? 

### Related Questions:
1. What are the conditions for this system to have no solutions?
2. What values of \( b \) make the system have infinitely many solutions?
3. How can we express the solution for \( x \) and \( y \) in terms of \( b \) and \( k \)?
4. What is the determinant of the coefficient matrix for this system?
5. How does changing \( k \) affect the solution?

**Tip:** If \( b = 3 \), the equations might be dependent, leading to special cases like infinitely many solutions or no solution.

Please solve the system of equations: {3x + 2y = 1} and {bx + 4y = k}.

Explore how to solve the system of equations 3x + 2y = 1 and bx + 4y = k using algebraic methods such as substitution and elimination. This guide provides step-by-step solutions and discusses the conditions under which the system has unique, infinite, or no solutions.

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