Finding G(n) A Step-by-Step Solution

Hey there, math enthusiasts! Ever stumbled upon a math problem that looks like a cryptic puzzle? Well, today, we're diving headfirst into one of those! We've got this equation: 3m + 5n = 11, and our mission, should we choose to accept it, is to find a formula for g(n) in terms of n. Basically, we need to figure out how to express 'm' (which is g(n)) using 'n'. Sounds like fun, right? Let's put on our thinking caps and get started!

Unraveling the Equation: A Step-by-Step Guide

So, the core of our challenge lies in the equation 3m + 5n = 11. Our primary goal here is to isolate 'm' on one side of the equation. Think of it like untangling a knot – we need to carefully maneuver the terms around until we have 'm' all by itself. The first step? Let's get rid of that '5n' term. We can do this by subtracting '5n' from both sides of the equation. Remember, in math, whatever you do to one side, you've gotta do to the other to keep things balanced. So, we have:

3m + 5n - 5n = 11 - 5n

This simplifies beautifully to:

3m = 11 - 5n

We're getting closer! Now, 'm' is almost free, but it's still clinging to that '3'. To completely isolate 'm', we need to divide both sides of the equation by 3. This will undo the multiplication and finally give us our formula. So, let's do it:

(3m) / 3 = (11 - 5n) / 3

And there we have it! Our final formula, shining in all its glory, is:

m = (11 - 5n) / 3

But wait, there's more! Remember that 'm' is actually g(n)? So, we can rewrite our formula as:

g(n) = (11 - 5n) / 3

This is the formula we've been searching for! We've successfully expressed g(n) in terms of n. This equation now tells us that for any input value 'n', we can plug it into this formula and get the corresponding output value 'm', which is g(n). We did it, guys! We cracked the code!

Deep Dive: Understanding the Formula and Its Implications

Now that we've nailed down the formula g(n) = (11 - 5n) / 3, let's take a moment to truly understand what it's telling us. This isn't just about crunching numbers; it's about seeing the bigger picture. This formula is a linear equation, which means that when we graph it, it will form a straight line. The 'n' is our input, the 'g(n)' is our output, and the formula itself defines the relationship between these two. For every value we plug in for 'n', we get a corresponding value for 'g(n)'.

But here's where it gets interesting. Notice that the formula involves division by 3. This means that not every value of 'n' will result in a whole number for g(n). To get a whole number for g(n), the expression '(11 - 5n)' must be divisible by 3. This adds a layer of complexity to our understanding. We can't just plug in any number for 'n' and expect a neat, integer answer for 'm'. This is a crucial aspect to consider, especially if our problem context requires whole number solutions.

Think of it like this: if we were dealing with a real-world scenario where 'm' and 'n' represent quantities of physical objects, we couldn't have fractional values. For example, if 'm' represented the number of apples and 'n' represented the number of oranges, we couldn't have 2.5 apples or 1.7 oranges. We can only have whole apples and whole oranges. In such cases, we'd need to find integer solutions for 'n' that make '(11 - 5n)' divisible by 3. To find those integer solutions, we might need to try different values of n or use some number theory concepts to narrow down the possibilities. Integer solutions can be found by plugging in values for n, such as if n = -2, g(n) = 7. Likewise, if n = 1, g(n) = 2. These points, (-2,7) and (1,2), will form solutions to the equation and can be plotted on the line if the integer requirements are required for the solution.

This highlights a critical skill in mathematics: not just finding a formula, but also interpreting its meaning and understanding its limitations. A formula is a powerful tool, but it's only as good as our understanding of it. So, always remember to think critically about the results you get and whether they make sense in the context of the problem.

Real-World Applications: Where Does This Formula Fit In?

You might be wondering,