To solve this puzzle involving two numbers, let's define two variables:

• Let the first number be $x$.
• Let the second number be $y$.

Step 1: Translate the clues into equations.

From the clues provided:

1. "Seven times the first number plus six times the second number equals 31."

   • This gives the equation:
     $7x + 6y = 31$

2. "Three times the first number minus ten times the second number is 29."

   • This gives the equation:
     $3x - 10y = 29$

Step 2: Solve the system of equations.

We now have the following system of equations: 7x + 6y = 31 \tag{1} 3x - 10y = 29 \tag{2}

Let's solve these equations simultaneously.

Solve for $x$ and $y$:

I'll calculate this next.The solution to the system of equations is:

$x = \frac{11}{2} = 5.5, \quad y = -\frac{5}{4} = -1.25$

Thus, the first number is $5.5$ and the second number is $-1.25$.

Would you like further details on how this was solved?

Here are 5 related questions you might explore:

1. What are some methods to solve a system of linear equations?
2. How do you graphically represent a system of equations and find their solution?
3. Can systems of linear equations have no solution or infinitely many solutions?
4. How can substitution and elimination methods be compared when solving systems of equations?
5. What role do determinants play in solving systems of equations using matrices?

Tip: Always check your solution by plugging the values back into the original equations to verify the correctness.