Solving 12x + 1 = 3(4x + 1) - 2: How Many Solutions?

Hey guys! Today, we're diving into a math problem that looks deceptively simple but can be quite interesting. We're going to figure out how many solutions exist for the equation 12x + 1 = 3(4x + 1) - 2. This kind of problem is a classic example of linear equations, and understanding how to solve them is super important for all sorts of math and real-world applications. Let's break it down step by step and see what we can discover!

Understanding the Equation

Before we jump into solving, let's take a good look at our equation: 12x + 1 = 3(4x + 1) - 2. Equations like this are called linear equations, and they involve variables (in this case, 'x') raised to the power of 1. The goal is to find the value(s) of 'x' that make the equation true. To do this, we need to simplify and rearrange the equation until we isolate 'x' on one side.

When we first glance at the equation, it seems like a standard linear equation. We've got 'x' terms, constants, and parentheses that we'll need to deal with. The key here is to follow the order of operations (PEMDAS/BODMAS) and carefully simplify each side of the equation. This involves distributing any multiplication over parentheses, combining like terms, and then strategically moving terms around to isolate 'x'. Sounds like a plan, right? Let’s dive deeper into how we'll tackle this!

First, we notice the parentheses on the right side of the equation. To get rid of those, we'll use the distributive property. This means multiplying the 3 outside the parentheses by each term inside: 4x and 1. Doing this will help us simplify the equation and make it easier to work with. Remember, the distributive property is a fundamental tool in algebra, and mastering it will help you solve all sorts of equations. So, let's put it into action here and see where it leads us.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this equation step-by-step. Remember our equation: 12x + 1 = 3(4x + 1) - 2.

1. Distribute

First, we need to get rid of those parentheses. We'll distribute the 3 on the right side of the equation:

12x + 1 = 3 * 4x + 3 * 1 - 2

This simplifies to:

12x + 1 = 12x + 3 - 2

2. Combine Like Terms

Now, let's combine the constant terms on the right side:

12x + 1 = 12x + 1

3. Analyze the Result

Whoa, check this out! We ended up with 12x + 1 = 12x + 1. Notice anything special about this? Both sides of the equation are exactly the same! This means that no matter what value we plug in for 'x', the equation will always be true. This is a crucial observation that tells us a lot about the solutions to this equation.

This situation is a bit different from what we usually encounter. Typically, when solving for 'x', we aim to isolate it on one side and find a specific value. But here, the 'x' terms canceled each other out, leaving us with an identity – a statement that's always true. This indicates that there isn't just one solution; instead, there's a whole range of possible solutions. Let's dig into what that means in the next section.

Infinite Solutions Explained

So, we've arrived at the equation 12x + 1 = 12x + 1. This is a special kind of equation called an identity. An identity is an equation that is true for all values of the variable. In our case, no matter what number we substitute for 'x', the left side of the equation will always equal the right side.

Think about it like this: if we try to solve for 'x' by subtracting 12x from both sides, we get:

12x + 1 - 12x = 12x + 1 - 12x

This simplifies to:

1 = 1

This statement is always true, regardless of 'x'. This confirms that there are infinitely many solutions. Any number you can think of will satisfy the original equation. This might seem a bit mind-bending, but it’s a perfectly valid outcome in algebra!

When we encounter an identity like this, it's a sign that the original equation represents a fundamental relationship. In geometric terms, if this were a linear equation graphed on a coordinate plane, the two sides of the equation would represent the same line. They completely overlap, meaning every point on the line is a solution. This is why we say there are infinitely many solutions – because there's no single value of 'x' that works; any value will do!

Contrast with Other Solution Types

It's important to understand that not all equations have infinitely many solutions. In fact, most linear equations we encounter have either one solution or no solutions at all. Let's take a quick look at these different scenarios to give you a broader perspective.

One Solution

Most linear equations have exactly one solution. For example, let's consider the equation 2x + 3 = 7. To solve this, we would subtract 3 from both sides:

2x = 4

Then, we divide by 2:

x = 2

Here, there's only one value of 'x' that makes the equation true: x = 2. If you plug in any other number for 'x', the equation won't balance.

No Solutions

Sometimes, an equation has no solution. This happens when we simplify the equation and end up with a contradiction – a statement that is always false. For instance, consider the equation 3x + 5 = 3x + 2. If we subtract 3x from both sides, we get:

5 = 2

This is clearly not true! No matter what value we substitute for 'x', the equation will never be balanced. In this case, we say the equation has no solutions.

Understanding these different solution types is essential for mastering algebra. Recognizing whether an equation has one solution, no solutions, or infinitely many solutions can save you a lot of time and effort. It also gives you a deeper insight into the relationships between variables and constants in an equation.

Real-World Implications

You might be wondering,