Solving 5^x + 5^y = 150 And X - Y = 1 A Mathematical Exploration

Hey there, math enthusiasts! Ever stumbled upon an equation that just makes you scratch your head? Well, today, we're diving deep into a fascinating mathematical puzzle: 5^x + 5^y = 150 and x - y = 1. This isn't your everyday algebra problem, guys. It's a blend of exponential functions and linear equations, and solving it requires a dash of cleverness and a sprinkle of algebraic manipulation. So, buckle up, because we're about to embark on a mathematical adventure!

Decoding the Equations: A Step-by-Step Journey

To begin our mathematical quest, let's first dissect the equations at hand. We have two equations, each presenting a unique challenge. The first equation, 5^x + 5^y = 150, is an exponential equation. The variables x and y appear as exponents, which means we're dealing with a different kind of beast compared to your typical linear equation. Exponential equations often require us to think outside the box, using logarithms or other special techniques to isolate the variables. On the other hand, the second equation, x - y = 1, is a linear equation. This one's a bit more familiar, isn't it? It tells us that the difference between x and y is 1, which gives us a crucial relationship between the two variables. This simple equation is our secret weapon, as it allows us to express one variable in terms of the other, simplifying the overall problem. The key to cracking this puzzle lies in recognizing how these two equations interact. We need to find a way to combine them, using the information from the linear equation to tame the exponential one. This is where the magic of substitution comes in, and we'll explore that in detail in the next section. Remember, guys, in mathematics, every equation is a piece of a larger puzzle, and our job is to fit the pieces together to reveal the complete picture. Let's move on and see how we can use substitution to make our lives easier!

The Power of Substitution: Taming the Exponential Beast

Alright, now that we've got a handle on our equations, let's talk strategy. The key to solving this system of equations lies in a technique called substitution. This is a fundamental tool in algebra, and it's particularly handy when dealing with systems of equations, like the one we have here. The basic idea behind substitution is simple: we use one equation to express one variable in terms of the other, and then we substitute that expression into the other equation. This effectively reduces the number of variables in the second equation, making it easier to solve. In our case, the linear equation x - y = 1 is perfectly suited for this. We can easily rearrange it to express x in terms of y, or vice versa. Let's choose to express x in terms of y. Adding y to both sides of the equation, we get x = y + 1. See how simple that was? Now we have a direct relationship between x and y. This is where the magic happens. We take this expression for x and substitute it into the exponential equation 5^x + 5^y = 150. This gives us 5^(y+1) + 5^y = 150. Notice what we've done here, guys? We've transformed the original equation, which had two variables, into a new equation with only one variable, y. This is a huge step forward! Now, we have a single equation that we can actually solve. But, before we jump into solving it, we need to do a little more algebraic maneuvering to make it even simpler. Remember, the goal is to isolate y, and to do that, we need to use the properties of exponents. Let's head on to the next section and see how we can simplify this equation further.

Simplifying the Equation: Unleashing the Exponent Rules

Okay, we've arrived at a crucial juncture in our mathematical journey. We've successfully substituted x = y + 1 into the exponential equation, and now we're staring at 5^(y+1) + 5^y = 150. At first glance, this might still look a bit intimidating, but don't worry, guys! We have a powerful weapon in our arsenal: the exponent rules. These rules are the key to unlocking the secrets of exponential expressions. One of the most fundamental exponent rules states that a^(m+n) = a^m * a^n. This rule allows us to break down exponents that are sums into products of exponents. In our case, we can apply this rule to the term 5^(y+1). We can rewrite it as 5^y * 5^1, which is simply 5^y * 5. Now, our equation looks like this: 5^y * 5 + 5^y = 150. See how much simpler it's becoming? We're making progress! The next step is to notice that we have a common factor of 5^y in both terms on the left side of the equation. This allows us to factor out 5^y, just like we would with any other common factor. Factoring out 5^y, we get 5^y (5 + 1) = 150, which simplifies to 5^y * 6 = 150. Wow, look at that! We've transformed the equation into something much more manageable. Now, we have 5^y multiplied by a constant. To isolate 5^y, we simply divide both sides of the equation by 6. This gives us 5^y = 150 / 6, which simplifies to 5^y = 25. We're almost there, guys! Now we have a simple exponential equation that we can solve directly. In the next section, we'll finally crack the code and find the value of y.

Cracking the Code: Finding the Value of y

We've reached the home stretch, guys! After all our algebraic maneuvering, we've arrived at the beautifully simple equation 5^y = 25. Now, the question is: what value of y makes this equation true? This is where our understanding of exponents comes into play. We need to think about what power we need to raise 5 to in order to get 25. Remember that 25 can be written as 5 * 5, which is 5^2. So, we have 5^y = 5^2. Now, this is a crucial point. If the bases are the same (in this case, both sides have a base of 5), then the exponents must be equal. Therefore, we can confidently conclude that y = 2. Boom! We've found the value of y. But hold on, our journey isn't over yet. We've only found one piece of the puzzle. Remember, we're solving a system of equations, which means we need to find the values of both x and y that satisfy both equations. We know that y = 2, but what about x? This is where our earlier work comes back into play. We know that x = y + 1. Now that we know y, we can simply substitute it into this equation to find x. Let's do that in the next section and complete our mathematical quest!

Completing the Puzzle: Solving for x and Verifying the Solution

Alright, we've successfully found that y = 2. Now, let's find the value of x. We know from our earlier substitution that x = y + 1. So, if we plug in y = 2, we get x = 2 + 1, which means x = 3. Fantastic! We've found a potential solution: x = 3 and y = 2. But, as any good mathematician knows, we're not done until we've verified our solution. We need to make sure that these values actually satisfy both of our original equations. Let's start with the first equation, 5^x + 5^y = 150. Plugging in x = 3 and y = 2, we get 5^3 + 5^2 = 150. Let's calculate this: 5^3 = 5 * 5 * 5 = 125 and 5^2 = 5 * 5 = 25. So, 125 + 25 = 150. Check! The first equation is satisfied. Now, let's check the second equation, x - y = 1. Plugging in x = 3 and y = 2, we get 3 - 2 = 1. Check! The second equation is also satisfied. We've done it, guys! We've successfully solved the system of equations. We've found that x = 3 and y = 2 is the solution that satisfies both equations. This was quite the mathematical journey, wasn't it? We started with a seemingly complex problem, but by breaking it down step by step, using techniques like substitution and exponent rules, we were able to unravel the mystery. In the next section, we'll recap our journey and highlight the key takeaways from this problem.

The Grand Finale: Reflecting on Our Mathematical Journey

Wow, what a ride, guys! We've successfully navigated the intricate world of exponential and linear equations and emerged victorious. Let's take a moment to reflect on our journey and highlight the key steps we took to solve the problem 5^x + 5^y = 150 and x - y = 1. First, we recognized the nature of the equations. We identified that we were dealing with a system of equations, one exponential and one linear. This understanding was crucial because it guided our strategy. Next, we employed the powerful technique of substitution. We used the linear equation x - y = 1 to express x in terms of y (x = y + 1). This allowed us to reduce the number of variables in the exponential equation, making it more manageable. Then, we substituted this expression for x into the exponential equation, resulting in 5^(y+1) + 5^y = 150. This was a pivotal step in simplifying the problem. After the substitution, we unleashed the power of exponent rules. We used the rule a^(m+n) = a^m * a^n to rewrite 5^(y+1) as 5^y * 5. This allowed us to factor out 5^y and further simplify the equation. By factoring and dividing, we isolated 5^y and arrived at the equation 5^y = 25. This was a major breakthrough! We then recognized that 25 is 5^2, and therefore, y = 2. We had found one piece of the puzzle! With y = 2, we went back to our substitution equation x = y + 1 and found that x = 3. Finally, and this is crucial, we verified our solution. We plugged x = 3 and y = 2 back into both original equations to make sure they were satisfied. And they were! Our solution, x = 3 and y = 2, was confirmed. This problem highlights the importance of several key mathematical concepts: understanding the nature of equations, using substitution to simplify problems, applying exponent rules effectively, and always verifying your solutions. These are valuable tools that will serve you well in your future mathematical adventures. So, the next time you encounter a challenging equation, remember the steps we took today, and you'll be well on your way to cracking the code! Keep exploring, keep learning, and keep enjoying the beauty of mathematics, guys!

Extra practice questions

1. Solve the system of equations: 2^x + 2^y = 20 and x + y = 5
2. Find the values of a and b that satisfy: 3^(a+1) + 3^b = 36 and a - b = 1
3. Determine the solution for: 4^m - 2^n = 112 and m - n = 2