Let's break down the problem:

We are given two conditions:

1. "The larger of two numbers is nine less than four times the smaller number."
2. "The sum of the two numbers is 76."

Let the smaller number be $x$, and the larger number be $y$.

From condition 1:

$y = 4x - 9$ (The larger number $y$ is 9 less than 4 times the smaller number $x$).

From condition 2:

$x + y = 76$ (The sum of the two numbers is 76).

Now, substitute the expression for $y$ from the first equation into the second equation:

$x + (4x - 9) = 76$ Simplifying: $5x - 9 = 76$ $5x = 85$ $x = 17$

Now substitute $x = 17$ back into the equation for $y$:

$y = 4(17) - 9 = 68 - 9 = 59$

Final answer:

The two numbers are $17$ and $59$.

Would you like further details on the solution? Here are 5 related questions to explore:

1. How would the solution change if the sum of the numbers was 80 instead of 76?
2. What if the larger number was 12 less than four times the smaller number?
3. How can we verify that our solution is correct?
4. Can you set up a similar system if the larger number was 7 more than twice the smaller number?
5. How would the process change if the larger number was expressed in terms of the sum, instead of a multiple of the smaller number?

Tip: When solving systems of equations, always double-check by substituting your solutions back into the original conditions.