Solving For 3x + 4y In A System Of Equations A Step By Step Guide
Have you ever encountered a system of equations that seemed a bit daunting? Don't worry, guys! Solving systems of equations is a fundamental skill in mathematics, and it's super useful in many real-world applications. In this article, we'll tackle one such system step by step. Our goal is to find the values of x and y from the given equations and then determine the value of the expression 3x + 4y. Let’s dive in and make math fun!
Understanding the Problem
The problem we're facing involves two equations:
- 4x = 5y + 13
- 3y = 7 - 5x
Our mission, should we choose to accept it (and we do!), is to find the values of x and y that satisfy both equations simultaneously. Once we have these values, we can easily compute 3x + 4y. Systems of equations like these pop up everywhere – from physics to economics – so mastering them is a fantastic skill to have. To start, we need a strategy. There are a few common methods for solving such systems, including substitution, elimination, and graphing. For this particular problem, we will use the substitution method, which involves solving one equation for one variable and then substituting that expression into the other equation. This transforms the system into a single equation with one variable, which is much easier to solve. So, let’s roll up our sleeves and get to work!
Choosing a Method: Substitution
So, why did we choose the substitution method? Good question! The substitution method shines when one of the equations can be easily solved for one variable in terms of the other. Looking at our equations:
- 4x = 5y + 13
- 3y = 7 - 5x
We can see that equation (2) is a great candidate for solving for y in terms of x. This is because we can isolate y on one side without too much hassle. The other popular method, elimination, involves adding or subtracting multiples of the equations to eliminate one variable. While elimination can be powerful, it might involve more steps in this particular case. Substitution allows us to reduce the problem to a single equation in one variable, which is often easier to handle. The graphing method, while visually appealing, isn't always the most precise for finding exact solutions, especially if the solutions aren't integers. Plus, graphing can be time-consuming. For these reasons, substitution is our method of choice here. Now that we’ve made our choice, let’s get into the nitty-gritty of the steps.
Step-by-Step Solution
Alright, guys, let's break this down step by step. Remember our equations:
- 4x = 5y + 13
- 3y = 7 - 5x
Step 1: Solve for One Variable
We decided to solve the second equation for y. So, let's isolate y in 3y = 7 - 5x. To do this, we simply divide both sides of the equation by 3:
y = (7 - 5x) / 3
Now we have y expressed in terms of x. This is crucial for the next step. Having y isolated sets us up perfectly for the substitution method. It's like having a key that unlocks the next part of the puzzle. Making sure we solve for the variable correctly is super important; any mistake here will throw off the rest of the solution. So, double-check your work and make sure everything looks good before moving on. Great job so far!
Step 2: Substitute
Now comes the fun part: substitution! We're going to take the expression we found for y, which is y = (7 - 5x) / 3, and plug it into the first equation. Remember the first equation? It's 4x = 5y + 13. So, wherever we see a y in the first equation, we're going to replace it with (7 - 5x) / 3. This gives us:
4x = 5 * ((7 - 5x) / 3) + 13
See what we did there? We've essentially swapped y for an expression involving x. This is the heart of the substitution method. We've transformed our two-variable system into a single equation with only x. This is a huge step forward because we can now solve for x. The equation might look a bit messy right now, but don't worry! We're going to clean it up in the next step. Just remember the basic principle: substitute the expression for one variable into the other equation. You’ve got this!
Step 3: Simplify and Solve for x
Okay, guys, let's tackle this equation: 4x = 5 * ((7 - 5x) / 3) + 13. It might look a bit intimidating, but we'll simplify it step by step. First, let's get rid of the fraction. We can do this by multiplying every term in the equation by 3. This gives us:
3 * (4x) = 3 * (5 * ((7 - 5x) / 3)) + 3 * 13
This simplifies to:
12x = 5(7 - 5x) + 39
Now, let's distribute the 5 across the parentheses:
12x = 35 - 25x + 39
Next, we combine like terms on the right side:
12x = 74 - 25x
Now, we want to get all the x terms on one side of the equation. So, let's add 25x to both sides:
12x + 25x = 74
This simplifies to:
37x = 74
Finally, to solve for x, we divide both sides by 37:
x = 74 / 37
So, we find that:
x = 2
Woohoo! We've found the value of x. That's a major milestone. Remember, math is like building blocks; we're using this value to find y in the next step. We’re on a roll, so let's keep going!
Step 4: Substitute x to Find y
Fantastic job on finding x, guys! Now that we know x = 2, we can use this value to find y. Remember the equation we found in Step 1, where we expressed y in terms of x? It was:
y = (7 - 5x) / 3
Now, we simply substitute x = 2 into this equation:
y = (7 - 5 * 2) / 3
Let's simplify:
y = (7 - 10) / 3
y = -3 / 3
So, we have:
y = -1
Excellent! We've found the value of y. Now we know both x and y. We're getting closer to the finish line. This step shows why substitution is so powerful; once we find one variable, plugging it back in is a breeze. We're on the home stretch now, with just one more step to go. Let's keep the momentum going and find the value of 3x + 4y.
Step 5: Calculate 3x + 4y
Alright, let's bring it all home! We've found that x = 2 and y = -1. The final part of the problem asks us to find the value of the expression 3x + 4y. This is a straightforward calculation. We simply substitute the values of x and y into the expression:
3x + 4y = 3 * 2 + 4 * (-1)
Now, let's do the math:
3x + 4y = 6 - 4
So, we get:
3x + 4y = 2
And there we have it! The value of 3x + 4y is 2. We've successfully solved the system of equations and answered the question. Give yourselves a pat on the back, guys. You've tackled a challenging problem step by step, and that's something to be proud of. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time.
Conclusion
Solving systems of equations is a crucial skill in mathematics, and we've just walked through one method – substitution – to tackle a specific problem. We started with the equations 4x = 5y + 13 and 3y = 7 - 5x and step by step, we found that x = 2 and y = -1. We then used these values to calculate 3x + 4y, which turned out to be 2. Remember, the key to mastering these problems is breaking them down into manageable steps. Whether it's isolating a variable, substituting expressions, or simplifying equations, each step brings you closer to the solution. Keep practicing, and you'll find these problems become second nature. Math can be fun, especially when you see how each step builds on the previous one to reveal the final answer. Keep exploring, keep learning, and remember, every problem is just a puzzle waiting to be solved!
