Math Problem: Find A+b+c If A-b=c And B+c=85

Let's dive into this interesting math problem! We are given that in an operation a - b = c, the sum of b and c is 85. Our mission, should we choose to accept it, is to find the sum of a, b, and c. Sounds like fun, right? So, let's break it down step by step and make sure we understand every little detail. Understanding the basics is super important, guys, or else we might end up scratching our heads later. Let's get started!

Understanding the Problem

Before we start solving, let's make sure we understand what the problem is asking. We have three variables: a, b, and c. Think of a as the starting number, b as the number we subtract from a, and c as the result we get after the subtraction. The problem gives us two key pieces of information:

1. a - b = c
2. b + c = 85

What we need to find is the value of a + b + c. Now that we have a clear picture of what we know and what we need to find, we can start thinking about how to connect the dots. Remember, in math, it's all about finding the right connections and using them to our advantage. So, let's put on our thinking caps and get ready to solve this puzzle!

Setting Up the Equations

We have two equations:

• Equation 1: a - b = c
• Equation 2: b + c = 85

Our goal is to find a + b + c. Notice that Equation 1 already gives us a relationship between a, b, and c. We can rewrite Equation 1 as a = b + c. This is a crucial step because it allows us to express a in terms of b and c. Now, we can use this new understanding to simplify our target expression, which is a + b + c.

Substituting and Simplifying

Now that we know a = b + c, we can substitute this into the expression we want to find, which is a + b + c. Replacing a with (b + c) gives us:

(b + c) + b + c

This looks much simpler, doesn't it? Now we can combine like terms. We have two b's and two c's, so we can rewrite the expression as:

2b + 2c

We can further simplify this by factoring out the 2:

2(b + c)

Using the Given Information

Remember that the problem told us that b + c = 85. This is exactly what we have inside the parentheses in our simplified expression! So, we can substitute 85 for (b + c):

2(85)

Now it's just a simple multiplication problem. What is 2 times 85? It's 170. So, we have found that:

a + b + c = 170

The Solution

Therefore, the sum of a, b, and c is 170. Awesome, right? We took a seemingly complex problem, broke it down into smaller, manageable steps, and used the given information to find the solution. This is a great example of how understanding the relationships between variables and using substitution can help us solve algebraic problems. Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them creatively.

Alternative Approach

Let's explore another way to think about this problem to reinforce our understanding. We start with the equations:

1. a - b = c
2. b + c = 85

We want to find a + b + c. We can rearrange the expression we want to find as:

a + (b + c)

Notice that we already know the value of (b + c) from the second equation. It's 85. So, we can rewrite the expression as:

a + 85

Now, we need to find the value of a. From the first equation, we have a - b = c. Adding b to both sides gives us:

a = b + c

But we know that b + c = 85, so:

a = 85

Now we can substitute this value of a back into our expression a + 85:

85 + 85

Which is equal to:

170

So, we arrive at the same answer: a + b + c = 170.

Key Takeaways

• Understand the Problem: Make sure you know what the problem is asking and what information you are given.
• Set Up Equations: Write down the given information as equations. This helps to visualize the relationships between the variables.
• Substitute and Simplify: Use substitution to eliminate variables and simplify the expressions.
• Use Given Information: Don't forget to use all the information provided in the problem. Sometimes, a seemingly small piece of information can be the key to solving the problem.
• Check Your Work: If possible, check your answer by plugging it back into the original equations.

Why This Matters

Understanding how to solve problems like this is important not just for math class, but also for developing critical thinking skills that can be applied in many areas of life. Breaking down complex problems into smaller steps, identifying key relationships, and using logical reasoning are valuable skills that can help you succeed in whatever you do. Plus, it's just plain fun to solve a good puzzle!

Keep practicing and you'll become a math whiz in no time! Solving problems like this is like exercising your brain – the more you do it, the stronger it gets. And remember, it's okay to make mistakes. Mistakes are just opportunities to learn and grow. So, don't be afraid to try, and don't give up! You got this!