Solving Integer Word Problems With Equations A Step By Step Guide

Hey guys! Let's dive into the fascinating world of algebra and tackle a classic word problem. These kinds of problems might seem tricky at first, but with a bit of practice and the right approach, you'll be solving them like a pro in no time. Today, we're going to break down a problem that involves finding two integers based on given relationships. Think of it like detective work with numbers – super cool, right?

Problem Statement

The problem we're tackling today goes like this: The larger of two integers is 8 less than twice the smaller. When the smaller number is subtracted from the larger, the difference is 17. Find the two numbers.

This might seem like a lot of information at once, but don't worry, we'll break it down piece by piece.

Step 1: Defining Variables

Okay, so the first thing we need to do when faced with a word problem like this is to translate the words into mathematical language. That means assigning variables to the unknowns. In our case, we have two unknown integers: a smaller one and a larger one. Let's make it simple and use:

• x to represent the smaller integer.
• y to represent the larger integer.

Why use variables? Well, variables are like placeholders. They allow us to represent unknown quantities and manipulate them using the rules of algebra. It's like giving our unknowns nicknames so we can talk about them more easily. Think of it this way: x and y are now our secret code for the two numbers we're trying to find.

Step 2: Translating Words into Equations

Now comes the fun part: turning the word problem into mathematical equations. This is where we take the relationships described in the problem and express them using our variables and mathematical symbols. Let's look at the two key pieces of information we have:

1. "The larger of two integers is 8 less than twice the smaller."

   Let's break this down: "twice the smaller" means 2 times the smaller integer, which we've called x. So that's 2x. Then, "8 less than" means we subtract 8. So, the entire phrase translates to 2x - 8. And the problem tells us this is equal to the larger integer, which we've called y. So our first equation is:

   y = 2x - 8

   See how we took a sentence and turned it into a neat little equation? This is the power of algebra!

2. "When the smaller number is subtracted from the larger, the difference is 17."

   Okay, this one's a bit more straightforward. "Smaller number subtracted from the larger" means y - x. And "the difference is 17" means equals 17. So our second equation is:

   y - x = 17

   Awesome! We've now transformed the word problem into two algebraic equations. This is a huge step, guys! We've taken a complicated verbal description and turned it into a system of equations that we can actually solve.

Step 3: Solving the System of Equations

We've got two equations and two unknowns – that's perfect! This means we can use a variety of methods to solve for x and y. One of the most common and effective methods is called substitution. The substitution method works by solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable and leaves us with a single equation in one variable, which we can easily solve.

Looking at our equations:

• y = 2x - 8
• y - x = 17

The first equation is already solved for y, which makes our job even easier! We can directly substitute the expression 2x - 8 for y in the second equation. This gives us:

(2x - 8) - x = 17

Now we have an equation with only x, and we can solve for it. First, let's simplify by combining like terms:

2x - x - 8 = 17

x - 8 = 17

Next, we add 8 to both sides of the equation to isolate x:

x - 8 + 8 = 17 + 8

x = 25

Yay! We've found x! Remember, x represents the smaller integer, so we now know the smaller integer is 25.

But we're not done yet! We still need to find y, the larger integer. This is where the substitution method really shines. Now that we know x = 25, we can substitute this value back into either of our original equations to solve for y. Let's use the first equation, since it's already solved for y:

y = 2 * 25 - 8

Now we just need to simplify:

y = 50 - 8

y = 42

Awesome! We've found y! The larger integer is 42.

Step 4: Checking the Solution

Before we celebrate our victory, it's always a good idea to check our solution. This ensures that our answers actually satisfy the conditions of the original problem. We can do this by plugging our values for x and y back into the original equations:

• y = 2x - 8
• y - x = 17

Let's substitute x = 25 and y = 42 into the first equation:

42 = 2 * 25 - 8

42 = 50 - 8

42 = 42

The first equation checks out! Now let's do the same for the second equation:

42 - 25 = 17

17 = 17

The second equation checks out too!

Since our solution satisfies both equations, we can be confident that we've found the correct integers.

Step 5: Stating the Answer

Finally, let's state our answer clearly and concisely. The problem asked us to find the two numbers, so we can say:

The smaller integer is 25, and the larger integer is 42.

That's it! We've successfully solved the word problem. Give yourselves a pat on the back, guys!

Key Takeaways and Strategies for Solving Integer Problems

Let's recap the key steps we took to solve this problem. These steps can be applied to a wide variety of integer problems:

1. Read the problem carefully: Understand what the problem is asking you to find and identify the key relationships between the unknowns.
2. Define variables: Assign variables to represent the unknown quantities. Choose variables that make sense in the context of the problem (like x for the smaller number and y for the larger number).
3. Translate words into equations: This is the heart of the problem-solving process. Break down the sentences into smaller phrases and translate them into mathematical expressions and equations. Look for keywords like "is," "less than," "more than," "difference," etc., as these often indicate mathematical operations.
4. Solve the system of equations: Use algebraic techniques like substitution or elimination to solve for the unknown variables. Choose the method that seems most efficient for the given equations.
5. Check the solution: Substitute your solutions back into the original equations to make sure they satisfy the conditions of the problem. This is a crucial step to avoid errors.
6. State the answer: Clearly and concisely state your answer in the context of the problem. Make sure you answer the question that was asked.

Extra Tips and Tricks:

• Practice makes perfect: The more word problems you solve, the better you'll become at translating them into equations.
• Draw diagrams or charts: Sometimes visualizing the problem can help you understand the relationships between the unknowns.
• Work backward: If you're stuck, try working backward from the answer choices (if provided) to see which ones satisfy the conditions of the problem.
• Don't be afraid to ask for help: If you're struggling, don't hesitate to ask your teacher, classmates, or a tutor for help. We all get stuck sometimes!

Solving word problems can be challenging, but it's also incredibly rewarding. It's like unlocking a secret code and revealing the hidden solution. Keep practicing, stay patient, and you'll become a master problem solver in no time!

Let's Practice!

Now that we've gone through an example problem step-by-step, let's try another one! This time, I encourage you to tackle it on your own, using the strategies we discussed. Here's a similar problem:

• The sum of two numbers is 50. The larger number is 10 more than twice the smaller number. Find the two numbers.

Give it a shot! Remember to define your variables, translate the words into equations, solve the system, check your solution, and state your answer clearly.

And that's a wrap, guys! Remember, math is like a muscle – the more you exercise it, the stronger it gets. Keep practicing, and you'll be amazed at what you can achieve!