Solving The System Of Equations: 3x – 4y = 10 And 5x – 8y = 22
Hey guys! Today, we're going to dive into solving a system of linear equations. Specifically, we're tackling the problem where we have two equations: 3x – 4y = 10 and 5x – 8y = 22. This might seem daunting at first, but trust me, with a little bit of algebra magic, we can crack this nut! We’ll explore the problem step by step, ensuring you understand every twist and turn in the solution process. So, let's roll up our sleeves and get started!
Understanding the Problem
Before we jump into solving, let's make sure we really understand what's going on. We have two equations, each with two variables, x and y. Our mission, should we choose to accept it (and we do!), is to find the values for x and y that make both equations true simultaneously. Think of it like finding the perfect meeting point for two lines on a graph. These types of problems are super common in algebra and have tons of real-world applications, from figuring out costs to planning projects. So, mastering this skill is seriously valuable.
Now, when dealing with these systems of equations, we have a couple of main methods at our disposal. The two most popular are the substitution method and the elimination method. Each has its strengths and can be more efficient depending on the specific equations we're working with. Today, we'll be focusing on the elimination method. Why? Because in this particular problem, it sets up quite nicely and can save us some steps. But don't worry, understanding both methods will make you a true equation-solving whiz! We'll break down the process into manageable steps so you can follow along easily. Let's get those variables figured out!
Choosing the Elimination Method
Okay, so we've decided to go with the elimination method for this problem. But why? Great question! The elimination method is particularly effective when we notice that the coefficients (the numbers in front of the variables) of either x or y are easy to manipulate to become the same or opposites. If we look closely at our equations:
- 3x – 4y = 10
- 5x – 8y = 22
We can see that the coefficients of y are -4 and -8. Notice anything interesting? That's right, -8 is a multiple of -4! This is a golden opportunity for elimination. By multiplying the first equation by 2, we can make the y coefficients match. This sets us up perfectly to eliminate y by subtracting one equation from the other. This strategic choice will simplify the system and allow us to solve for x relatively quickly.
The beauty of the elimination method is that it lets us neatly get rid of one variable, making the problem much easier to handle. In contrast, if we were to use substitution, we might end up dealing with fractions or more complex expressions, which can increase the chances of making a mistake. So, by recognizing the relationship between the y coefficients, we're making a smart move that will streamline our solution process. Keep your eyes peeled for these kinds of opportunities when you're solving systems of equations – they can save you a lot of time and effort!
Step-by-Step Solution
Alright, let’s dive into the step-by-step solution using the elimination method. We've already identified that manipulating the y coefficients is our best bet. Here's how we'll do it:
Step 1: Multiply the First Equation by 2
Our goal here is to make the coefficient of y in the first equation match the coefficient of y in the second equation. So, we'll multiply every term in the first equation (3x – 4y = 10) by 2:
2 * (3x – 4y) = 2 * 10
This gives us a new equation:
6x – 8y = 20
Now, we have a modified first equation that's ready to play nicely with our second equation.
Step 2: Write the Equations Together
Let's line up our modified first equation and the second equation so we can see them clearly:
- 6x – 8y = 20
- 5x – 8y = 22
Having them side-by-side makes it easier to visualize our next move, which is to eliminate the y variable.
Step 3: Subtract the Second Equation from the Modified First Equation
Since the coefficients of y are the same (-8), we can eliminate y by subtracting the second equation from the first. This means we'll subtract the left side of the second equation from the left side of the modified first equation, and do the same for the right sides:
(6x – 8y) – (5x – 8y) = 20 – 22
Now, let's simplify. Be careful with the signs!
6x – 8y – 5x + 8y = -2
Notice how the -8y and +8y cancel each other out? That's the magic of the elimination method!
This leaves us with:
x = -2
Woohoo! We've found the value of x! Now we're halfway there.
Step 4: Substitute the Value of x into One of the Original Equations
To find the value of y, we'll take the value of x we just found (-2) and plug it into one of our original equations. It doesn't matter which one we choose; we'll get the same answer either way. For this example, let's use the first original equation:
3x – 4y = 10
Substitute x = -2:
3(-2) – 4y = 10
Step 5: Solve for y
Now we have an equation with just one variable, y. Let's solve for it:
-6 – 4y = 10
Add 6 to both sides:
-4y = 16
Divide both sides by -4:
y = -4
And there we have it! We've found the value of y.
Step 6: Write the Solution as an Ordered Pair
Finally, let's write our solution as an ordered pair (x, y):
(-2, -4)
This ordered pair represents the point where the two lines intersect on a graph. It's the solution that satisfies both equations simultaneously. We’ve conquered this system of equations!
Verification of the Solution
Before we celebrate our victory, it’s always a good idea to double-check our answer. Verification is like the safety net of algebra – it ensures we haven't made any sneaky errors along the way. To verify, we'll take our solution, x = -2 and y = -4, and plug it back into both of the original equations. If our solution is correct, it should make both equations true.
Verification with the First Equation: 3x – 4y = 10
Substitute x = -2 and y = -4:
3(-2) – 4(-4) = 10
Simplify:
-6 + 16 = 10
10 = 10
Hooray! The first equation checks out.
Verification with the Second Equation: 5x – 8y = 22
Now, let's do the same with the second equation:
5(-2) – 8(-4) = 22
Simplify:
-10 + 32 = 22
22 = 22
Yes! The second equation also holds true. Since our solution satisfies both equations, we can confidently say that our answer is correct. We’ve not only solved the problem but also verified our work, making our solution rock-solid. This verification step is crucial, especially in exams or when dealing with more complex problems. It gives you peace of mind knowing that you've nailed it.
Conclusion
Alright guys, we did it! We successfully solved the system of equations 3x – 4y = 10 and 5x – 8y = 22 using the elimination method. We found that x = -2 and y = -4, which we wrote as the ordered pair (-2, -4). We also took the extra step of verifying our solution by plugging it back into both original equations, just to be sure we were spot-on. Remember, verification is your best friend in algebra!
Systems of equations might seem intimidating at first, but with a systematic approach and a little practice, you can conquer them. The elimination method is a powerful tool in your algebraic arsenal, especially when you spot those coefficients that are begging to be manipulated. Keep an eye out for those opportunities, and you'll be solving systems of equations like a pro in no time. And if you ever get stuck, remember to break the problem down into smaller, manageable steps, just like we did today. Happy solving! ✨
