Solving Equations How Many Solutions Exist For 1/2(x+12) = 4x - 1

Hey there, math enthusiasts! Ever stumbled upon an equation and wondered how many solutions it has? Well, you're in the right place. Today, we're diving deep into solving the equation $\frac{1}{2}(x+12)=4x-1$ and figuring out whether it has zero, one, two, or infinitely many solutions. So, grab your calculators and let's get started!

Understanding the Basics of Equations

Before we jump into the specifics, let's quickly recap what we mean by the solutions of an equation. Essentially, a solution is a value (or values) that, when plugged in for the variable (in this case, 'x'), makes the equation true. Think of it like a puzzle piece that perfectly fits into the equation's framework. If we substitute a value for 'x' and both sides of the equation are equal, then bingo! We've found a solution.

Now, equations can behave in different ways. Some equations, like the one we're tackling today, have a single, unique solution. Others might have no solutions at all, meaning there's no value for 'x' that can make the equation true. And then there are those equations that have infinitely many solutions, where any value for 'x' will satisfy the equation. Understanding these possibilities is key to mastering algebra.

Linear equations, like the one we're working with, are a fundamental part of algebra. They represent a straight line when graphed, and their solutions correspond to the points where the line intersects the x-axis (if we were to set the equation equal to zero). The number of solutions a linear equation has depends on its structure and the relationship between the variables and constants involved.

So, why is it important to know how many solutions an equation has? Well, in real-world applications, equations model various scenarios, from calculating distances and speeds to predicting financial outcomes. Knowing whether an equation has a solution, and how many, helps us make informed decisions and predictions. For example, in engineering, you might need to solve an equation to determine the optimal dimensions of a structure. If the equation has no solution, it means your design needs tweaking!

Step-by-Step Solution of the Equation

Alright, let's roll up our sleeves and solve the equation $\frac{1}{2}(x+12)=4x-1$ step-by-step. Don't worry, we'll break it down into manageable chunks. Our goal here is to isolate 'x' on one side of the equation, so we can see what value (or values) make the equation true.

Step 1: Distribute the $\frac{1}{2}$

First things first, we need to get rid of those parentheses. To do this, we'll distribute the $\frac{1}{2}$ across the terms inside the parentheses: $\frac{1}{2} * x$ and $\frac{1}{2} * 12$. This gives us:

$\frac{1}{2}x + 6 = 4x - 1$

See? We've already simplified the equation quite a bit. Distributing terms is a crucial step in solving many algebraic equations, so it's a good technique to have in your toolkit.

Step 2: Get Rid of the Fraction

Fractions can sometimes make things look messier than they are. So, let's eliminate the fraction by multiplying every term in the equation by 2. This will clear the fraction and make our equation easier to work with:

$2 * (\frac{1}{2}x + 6) = 2 * (4x - 1)$

This simplifies to:

$x + 12 = 8x - 2$

Much cleaner, right? Multiplying by a common denominator (in this case, 2) is a handy trick for dealing with equations involving fractions.

Step 3: Group the 'x' Terms

Now, let's gather all the 'x' terms on one side of the equation. We can do this by subtracting 'x' from both sides:

$x + 12 - x = 8x - 2 - x$

Which simplifies to:

$12 = 7x - 2$

Great! We're getting closer to isolating 'x'. Grouping like terms is a fundamental step in solving equations, ensuring we keep everything organized.

Step 4: Isolate the 'x' Term

Next, we want to get the term with 'x' by itself. To do this, we'll add 2 to both sides of the equation:

$12 + 2 = 7x - 2 + 2$

This gives us:

$14 = 7x$

We're almost there! Just one more step to go.

Step 5: Solve for 'x'

Finally, to solve for 'x', we'll divide both sides of the equation by 7:

$\frac{14}{7} = \frac{7x}{7}$

This simplifies to:

$x = 2$

Woo-hoo! We've found the solution. So, after all that algebraic maneuvering, we've determined that x = 2 is the single, unique solution to the equation.

Determining the Number of Solutions

Now that we've solved the equation and found that x = 2, we can confidently say that there is one solution. But, how can we tell, just by looking at the equation, whether it will have one solution, no solutions, or infinitely many solutions? Let's explore that a bit.

One Solution

As we saw in our step-by-step solution, the equation $\frac{1}{2}(x+12)=4x-1$ simplifies to a form where we can isolate 'x' and find a single value that satisfies the equation. This is typical of linear equations where the variable 'x' appears to the first power and there are no contradictions in the equation. When you solve the equation, you arrive at a unique value for 'x', indicating one solution.

In graphical terms, a linear equation represents a straight line. An equation with one solution means that if you were to graph the equation (or the two expressions on either side of the equation as separate lines), the lines would intersect at exactly one point. The x-coordinate of that point is the solution to the equation.

No Solutions

Sometimes, when you try to solve an equation, you might encounter a contradiction. For example, you might end up with an equation like 0 = 5, which is clearly not true. This indicates that the equation has no solutions. No matter what value you substitute for 'x', the equation will never be true.

Graphically, an equation with no solutions represents parallel lines. Parallel lines never intersect, so there's no point (no value of 'x') that satisfies both sides of the equation simultaneously.

Infinitely Many Solutions

On the flip side, some equations are true no matter what value you substitute for 'x'. These equations have infinitely many solutions. You'll often encounter this situation when the equation simplifies to an identity, like 0 = 0 or x = x. In these cases, any value for 'x' will make the equation true.

Graphically, an equation with infinitely many solutions represents the same line. The two expressions on either side of the equation are essentially different forms of the same line, so they overlap completely. Every point on the line is a solution to the equation.

Why is This Important?

Understanding how many solutions an equation has is not just a theoretical exercise. It has practical implications in various fields, from engineering and physics to economics and computer science. Equations are used to model real-world phenomena, and the number of solutions can tell us a lot about the system being modeled.

For example, in engineering, you might use equations to design a bridge or a building. The solutions to these equations might represent the forces acting on the structure. If the equations have no solutions, it could indicate a flaw in the design. If they have infinitely many solutions, it might mean the design is over-constrained.

In economics, equations are used to model supply and demand, predict market trends, and analyze financial data. The number of solutions can provide insights into the stability of the market and the potential for growth or recession.

In computer science, equations are used in algorithms for optimization, machine learning, and data analysis. Understanding the nature of the solutions is crucial for developing efficient and accurate algorithms.

So, the next time you encounter an equation, remember that the number of solutions is a key piece of information that can unlock deeper insights into the problem at hand.

Conclusion

So, to wrap things up, we've successfully solved the equation $\frac{1}{2}(x+12)=4x-1$ and determined that it has one solution, which is x = 2. We also explored the different types of solution sets equations can have – one solution, no solutions, and infinitely many solutions – and how to identify them. And, more importantly, we discussed why understanding the number of solutions is crucial in various real-world applications.

Keep practicing, keep exploring, and you'll become a math whiz in no time! Until next time, happy solving, folks!