Solving Equations: Find The Number Of Solutions!

Hey math enthusiasts! Let's dive into the world of equations and figure out how to determine the number of solutions a particular equation has. We'll tackle the equation $\frac{1}{2}(x+12)=4 x-1$ together, step by step, and see if it has zero, one, two, or infinitely many solutions. This is a crucial skill in algebra, so understanding how to solve it is key. So, grab your pencils and let's get started!

Understanding the Basics: Equations and Solutions

Alright, before we jump into the equation, let's quickly recap what an equation and its solutions are all about. In simple terms, an equation is a mathematical statement that asserts the equality of two expressions. Think of it like a balanced scale; both sides must be equal for the equation to hold true. The expressions are connected with an equals sign (=). For instance, something like $2 + 2 = 4$ is a simple equation.

Now, a solution to an equation is a value (or values) that, when substituted into the equation, makes the statement true. This value is usually represented by a variable, like 'x'. So, when we 'solve' an equation, we are essentially trying to find the value(s) of the variable that satisfy the equation. If we find a value that makes the equation true, we've found a solution. If we don't, then there's no solution or infinitely many.

For example, in the equation $x + 3 = 5$, the solution is $x = 2$ because when we substitute 2 for x, the equation becomes $2 + 3 = 5$, which is true. Got it? Awesome! Let's move on to the types of solutions we can encounter. There are primarily three types: one solution, no solution, or infinitely many solutions. We will explore each scenario as we go through the process.

Let's Solve the Equation: Step-by-Step Guide

Now, let's get down to the business of solving the equation $\frac{1}{2}(x+12)=4 x-1$. Here's how we'll do it, step by step: The goal here is to isolate 'x' on one side of the equation and figure out its value. This value, if we find one, is our solution. Are you ready?

Step 1: Distribute on the Left Side. First, we need to get rid of those parentheses. To do this, we distribute the $\frac{1}{2}$ across both terms inside the parentheses. So, $\frac{1}{2} * x$ becomes $\frac{1}{2}x$ and $\frac{1}{2} * 12$ becomes $6$. Our equation now looks like this: $\frac{1}{2}x + 6 = 4x - 1$.

Step 2: Get the x Terms Together. Next, we want to get all the terms with 'x' on one side of the equation and the constant terms (the numbers without 'x') on the other side. Let's subtract $\frac{1}{2}x$ from both sides. This gives us: $6 = 4x - \frac{1}{2}x - 1$. Simplifying the right side, we get $6 = 3.5x - 1$.

Step 3: Isolate the x Term. Now, let's add 1 to both sides to isolate the term with 'x'. This gives us $7 = 3.5x$.

Step 4: Solve for x. Finally, to solve for 'x', we divide both sides by 3.5. So, $x = \frac{7}{3.5}$, which simplifies to $x = 2$. Bingo! We've found a solution.

Analyzing the Solution and Answering the Question

Great job! We have solved the equation $\frac{1}{2}(x+12)=4 x-1$ and found that $x = 2$. This means there is only one value of 'x' that makes the equation true. Therefore, the equation has one solution. So, the correct answer is B. one. But what if we didn't get a single value for x? Let's talk about the different scenarios to fully understand this concept.

Now, let's consider the other options and what they would mean. If we had ended up with an equation that simplifies to something like $0 = 5$ or $2 = 8$, this would be a contradiction, indicating that there is no solution (Option A). This happens because there is no value for 'x' that will make the equation true. No matter what number we substitute for 'x', the equation will always be false.

On the other hand, if we had arrived at an equation that simplifies to something like $x = x$ or $5 = 5$, this means the equation is true for any value of 'x', and there are infinitely many solutions (Option D). This occurs when both sides of the equation are essentially identical, meaning any value of the variable will satisfy the equation. This is rarer, but it's important to recognize these different outcomes.

Different Types of Solutions

Let's go through the different possibilities of the solutions in a clearer way. We have already mentioned them but here we will make it explicit.

• One Solution: This is the most common scenario. When we solve an equation, we isolate the variable and get a single numerical value. In our example, we found $x = 2$. This means there is only one value that makes the equation true.
• No Solution: Sometimes, when solving an equation, we arrive at a contradiction. For example, if we simplified an equation to $5 = 0$. This statement is never true. In this case, there is no value of the variable that would make the original equation true. The lines would be parallel and never intersect.
• Infinitely Many Solutions: In this scenario, when we simplify the equation, we end up with an identity. For example, $x = x$ or $7 = 7$. This means the equation is true for any value of the variable. Both sides of the equation are essentially the same, indicating that any value substituted for the variable will satisfy the equation. The lines would be identical, overlapping at all points.

Tips and Tricks for Solving Equations

To make solving equations easier, here are some helpful tips and tricks. These tips will help you not only with this type of problem, but with many other types of mathematical problems as well.

• Simplify First: Always try to simplify both sides of the equation as much as possible before starting to move terms around. This includes combining like terms and removing parentheses. This makes the equation less cluttered and easier to solve.
• Keep Track of Signs: Be very careful with positive and negative signs. A small mistake can lead to a completely different answer. Double-check your work, especially when multiplying or dividing by negative numbers.
• Check Your Answer: After you think you've found a solution, plug it back into the original equation to verify that it works. This helps catch any calculation errors and builds confidence in your answer.
• Practice, Practice, Practice: The more you solve equations, the better you will become. Try different types of equations, including those with fractions, decimals, and variables on both sides. Practice makes perfect. Don't be afraid to make mistakes; they are a great way to learn!
• Use Visual Aids: If you're a visual learner, try drawing diagrams or using number lines to visualize the equation. This can help you understand the problem better.
• Understand the Concepts: Make sure you understand the underlying mathematical concepts, such as the order of operations, properties of equality, and how to manipulate equations. This will help you solve problems more effectively.
• Break Down Complex Problems: If an equation seems too complicated, break it down into smaller, more manageable steps. Solve each part separately and then combine the results.

Conclusion: Mastering Equation Solutions

So, there you have it! We've successfully solved the equation $\frac{1}{2}(x+12)=4 x-1$, determined that it has one solution, and explored the other possibilities of having no solutions or infinitely many solutions. Remember, understanding how to solve equations and analyze the types of solutions is a fundamental skill in algebra and beyond.

By following the steps and tips we discussed, you'll be well on your way to mastering these types of problems. Keep practicing, stay curious, and you'll find that solving equations can be a rewarding and enjoyable experience. Now, go forth and conquer those equations! Don't hesitate to revisit these concepts as you continue your mathematical journey. Happy solving!