Solving Math Problems Finding Two Numbers With Equations

Hey guys! Let's dive into a math problem that might seem a bit tricky at first, but don't worry, we'll break it down together. We're going to figure out how to find two numbers when we're given some clues about their sum and difference, involving multiples of the numbers. It’s like a little puzzle, and who doesn’t love solving a puzzle?

Understanding the Problem

Before we jump into solving, let's make sure we really understand what's being asked. The problem states: "The sum of a number with the triple of another is 18; and the difference of the first with the double of the second is 8. What are those numbers?"

So, what does this actually mean? Well, we have two unknown numbers. Let's call them 'x' and 'y' just to make things easier. The problem then gives us two pieces of information:

1. The sum of x and three times y is 18. We can write this as an equation: x + 3y = 18
2. The difference between x and two times y is 8. This translates to another equation: x - 2y = 8

See? It's like translating a secret code! Now we have two equations, and with two equations and two unknowns, we're in business. This is a classic system of equations problem.

Setting up the Equations: The Foundation of Our Solution

In this section, we're going to really solidify how we transform the word problem into mathematical equations. This is super important because if the equations aren't right, the answer won't be either. Think of it like building a house – the foundation needs to be solid!

Let’s reiterate those key phrases from the problem and how they become equations. We identified our two unknown numbers, and we've lovingly named them 'x' and 'y'. Remember, 'x' represents the first number, and 'y' is the second number. Now, let's break it down:

• "The sum of a number with the triple of another is 18" – This is where careful reading is crucial. The phrase "the sum of" tells us we're adding things together. "A number" refers to our 'x'. "The triple of another" means three times our 'y', which is 3y. And "is 18" simply means equals 18. So, we get our first equation: x + 3y = 18. This equation represents a relationship between our two numbers where if we add 'x' to three times 'y', we always end up with 18. It's like a balancing act between 'x' and 'y'.
• "The difference of the first with the double of the second is 8" – "The difference" indicates subtraction. "The first" is our 'x' again. "The double of the second" means two times 'y', or 2y. And "is 8" means equals 8. So, our second equation is: x - 2y = 8. This equation gives us another relationship – the value we get when we subtract twice 'y' from 'x' is always 8. This provides a different constraint on the possible values of 'x' and 'y'.

Now, we have a system of two linear equations:

x + 3y = 18 x - 2y = 8

These equations are the mathematical representation of our word problem. They're the key to unlocking the values of 'x' and 'y'. We've successfully translated the words into a language that we can use to solve the problem. High five!

Solving the System of Equations: Methods to Find the Numbers

Alright, now for the fun part – actually solving for 'x' and 'y'! There are a couple of main ways we can tackle this, and we'll go through one method in detail here: the elimination method. This method is particularly useful when the coefficients (the numbers in front of our variables) are the same or easy to make the same. Let's see how it works.

The Elimination Method: A Step-by-Step Guide

The goal of the elimination method is to get rid of one of the variables by adding or subtracting the equations. When we eliminate one variable, we're left with a single equation with a single unknown, which is much easier to solve. In our case, notice that both equations have 'x' with a coefficient of 1. This is perfect for elimination!

1. Line up the Equations: Make sure the 'x' terms, 'y' terms, and the constants are lined up:

x + 3y = 18 x - 2y = 8

2. Eliminate a Variable: Since the 'x' terms have the same coefficient, we can subtract the second equation from the first equation. This will eliminate 'x':

(x + 3y) - (x - 2y) = 18 - 8

Notice how we're subtracting the entire second equation. Be careful with the signs!

3. Simplify: Let's simplify the equation:

x + 3y - x + 2y = 10 The 'x' terms cancel out (x - x = 0), which is exactly what we wanted! 5y = 10

4. Solve for the Remaining Variable: Now we have a simple equation with just 'y'. Divide both sides by 5 to isolate 'y':

y = 10 / 5 y = 2

Yay! We've found one of our numbers: y = 2.

5. Substitute to Find the Other Variable: Now that we know 'y', we can plug it back into either of our original equations to solve for 'x'. Let's use the first equation, x + 3y = 18:

x + 3(2) = 18 x + 6 = 18 Subtract 6 from both sides: x = 18 - 6 x = 12

Awesome! We've found the other number: x = 12.

So, we've successfully solved the system of equations using the elimination method. It's like a little dance – we eliminated one variable, found the other, and then brought the first one back into the picture. Pretty cool, huh?

Checking Our Solution: Ensuring Accuracy

Before we declare victory, it's super important to check our solution. We want to make sure the numbers we found, x = 12 and y = 2, actually satisfy both of the original equations. This is our way of making sure we didn't make any sneaky mistakes along the way. Think of it as proofreading your work before you submit it!

Let's go back to our original equations:

1. x + 3y = 18
2. x - 2y = 8

Now, we'll substitute x = 12 and y = 2 into each equation:

Equation 1: x + 3y = 18

12 + 3(2) = 18 12 + 6 = 18 18 = 18

Woohoo! The first equation checks out. Our numbers make the equation true.

Equation 2: x - 2y = 8

12 - 2(2) = 8 12 - 4 = 8 8 = 8

Double woohoo! The second equation also checks out. Our numbers make this equation true as well.

Since our solution (x = 12, y = 2) satisfies both equations, we can confidently say that we've found the correct answer. It's like getting a thumbs up from the math gods!

The Answer and Its Significance: What We've Discovered

So, after all that solving, what's the final answer? We found that x = 12 and y = 2. This means the two numbers we were looking for are 12 and 2!

But it's not just about getting the right numbers; it's also about understanding what those numbers represent in the context of the original problem. Remember, the problem asked us to find two numbers that satisfy two specific conditions: their sum with the triple of the other being 18, and their difference with the double of the other being 8. We've successfully found those numbers.

The solution (12, 2) is unique. There are no other pairs of numbers that would satisfy both of these conditions simultaneously. This highlights the power of systems of equations – they allow us to find specific solutions to problems with multiple constraints.

Connecting to Real-World Scenarios: Why This Matters

You might be thinking, "Okay, cool, we solved a math problem. But when am I ever going to use this in real life?" Well, the truth is, systems of equations pop up in all sorts of unexpected places! They're not just abstract math concepts; they're tools for modeling and solving real-world situations.

Here are a few examples:

• Business and Finance: Imagine you're running a small business and you need to figure out how many of each product you need to sell to break even. You might have equations representing your costs, revenue, and profit. Solving those equations would help you determine your sales targets.
• Science and Engineering: Systems of equations are used extensively in physics, chemistry, and engineering to model and analyze systems. For example, electrical circuits can be analyzed using systems of equations to determine currents and voltages.
• Economics: Economic models often involve systems of equations that describe relationships between supply, demand, prices, and other factors.
• Everyday Life: Even in everyday situations, you might implicitly use systems of equations. For example, if you're trying to figure out how many hours to work at two different jobs to reach a certain income goal, you're essentially dealing with a system of equations.

The key takeaway here is that the skills we've used to solve this problem – setting up equations, using methods like elimination, and checking our solutions – are valuable skills that can be applied in many different fields. It's not just about the numbers; it's about the problem-solving process!

Practice Makes Perfect: Further Exploration

So, we've conquered this problem together! But like any skill, solving systems of equations gets easier with practice. Here are a few things you can do to keep honing your skills:

• Find similar problems: Look for other word problems that can be translated into systems of equations. Textbooks, online resources, and even puzzles can be great sources of practice problems.
• Try different methods: We focused on the elimination method here, but there's also the substitution method. Try solving the same problems using different methods to see which one you prefer and which one is most efficient for different types of problems.
• Create your own problems: This is a fun way to really solidify your understanding. Try coming up with your own scenarios that can be modeled with systems of equations.
• Work with a friend: Solving math problems with a friend can make the process more enjoyable and help you learn from each other.

Keep practicing, keep exploring, and keep challenging yourself! Math is a journey, and every problem you solve is a step forward.

Conclusion: You've Got This!

We've successfully solved a system of equations problem, and hopefully, you now feel a bit more confident in your ability to tackle similar challenges. Remember, the key is to break down the problem into smaller, manageable steps, translate the words into equations, choose an appropriate solution method, and always check your work.

Keep up the great work, and never stop exploring the fascinating world of mathematics! You've got this!