We are tasked with solving a simultaneous equation. The given system is:

\frac{7}{2x + 1} + \frac{13}{y + 2} = 27 \tag{1} \frac{13}{2x + 1} + \frac{7}{y + 2} = 33 \tag{2}

To solve these equations, we can follow these steps:

1. Set common terms: Define new variables for the denominators:

   • Let $A = \frac{1}{2x + 1}$
   • Let $B = \frac{1}{y + 2}$

   The system becomes: 7A + 13B = 27 \tag{3} 13A + 7B = 33 \tag{4}

2. Solve the linear system: We'll now solve these two equations simultaneously.

Let's work through this step-by-step. I'll calculate the solutions.The solution to the system of equations is:

$A = 2 \quad \text{and} \quad B = 1$

Now, recall the definitions of $A$ and $B$:

• $A = \frac{1}{2x + 1}$
• $B = \frac{1}{y + 2}$

Substituting the values of $A$ and $B$ back into these equations:

$\frac{1}{2x + 1} = 2 \quad \Rightarrow \quad 2x + 1 = \frac{1}{2} \quad \Rightarrow \quad x = -\frac{1}{4}$

$\frac{1}{y + 2} = 1 \quad \Rightarrow \quad y + 2 = 1 \quad \Rightarrow \quad y = -1$

Therefore, the solution to the original simultaneous equations is:

$x = -\frac{1}{4} \quad \text{and} \quad y = -1$

Would you like any further details or have any questions?

Here are some related problems to explore:

1. What happens if we modify the coefficients in the system of equations?
2. How do we solve this system if the equations were nonlinear?
3. Can we solve the system using substitution instead of elimination?
4. How can matrices be used to solve such a system of equations?
5. What are the graphical interpretations of these solutions?

Tip: When solving simultaneous equations, always look for patterns that allow simplifications, such as common denominators or variable elimination.