Unveiling Equation Solutions: Zero, One, Or Many?

Hey everyone! Today, we're diving into the fascinating world of equations and their solutions. Specifically, we're tackling the equation: $0.75(x+40)=0.35(x+20)+0.35(x+20)$. Our mission? To figure out just how many solutions this equation holds. Is it zero, one, two, or an infinite number? Let's break it down and find out! This is a core concept in mathematics, touching upon algebra, problem-solving, and understanding the nature of equations. Understanding the number of solutions is crucial because it tells us about the equation's behavior and the possible values of the variable that satisfy the equation. This particular equation is a linear equation, so we should anticipate finding either one solution, no solution, or infinitely many solutions. We will explore each scenario and eliminate the unlikely options.

Decoding the Equation: A Step-by-Step Approach

Alright, guys, let's roll up our sleeves and solve this equation step by step. Our goal is to isolate 'x' and find its value(s). By following the rules of algebra, we can manipulate the equation until we get 'x' all by itself on one side. Remember, the key to solving equations is to perform the same operation on both sides to maintain the balance. This ensures that the equation remains valid throughout the solving process. Let's start by simplifying the right side of the equation. We have two identical terms, 0.35(x + 20), which we can combine:

$$0.75(x + 40) = 0.35(x + 20) + 0.35(x + 20)$$

$$0.75(x + 40) = 2 * 0.35(x + 20)$$

$$0.75(x + 40) = 0.70(x + 20)$$

Now, let's distribute the numbers outside the parentheses:

$$0. 75x + 30 = 0.70x + 14$$

Next, let's get all the 'x' terms on one side and the constants on the other. We'll subtract 0.70x from both sides:

$$0.75x - 0.70x + 30 = 0.70x - 0.70x + 14$$

$$0.05x + 30 = 14$$

Then, subtract 30 from both sides:

$$0.05x + 30 - 30 = 14 - 30$$

$$0.05x = -16$$

Finally, divide both sides by 0.05 to solve for x:

$$x = -16 / 0.05$$

$$x = -320$$

So, there you have it! We've found that x = -320. This indicates that there is one and only one solution for this equation.

Why Solving Step-by-Step Matters

Solving equations step-by-step is super important, guys! It helps us avoid mistakes and makes sure we're following the right rules of algebra. When you write down each step, it's easier to spot errors and understand where things went wrong if you get a different answer. Plus, breaking down the problem into smaller parts makes it less overwhelming. By practicing this method, you become more confident in your abilities to solve any equation that comes your way. It's like building a solid foundation – each step is a brick that supports the entire structure of the solution. Keep practicing and you will get better!

Examining the Solution: Does it Make Sense?

Now that we've crunched the numbers and found our solution, let's check our work. Does x = -320 actually satisfy the original equation? This step is critical because it helps us verify that our calculations are correct. Let's plug -320 back into the equation and see if both sides are equal. This is known as checking our solution. It's about ensuring accuracy, which is essential in math. Let's do it!

$$0.75(-320 + 40) = 0.35(-320 + 20) + 0.35(-320 + 20)$$

First, simplify the terms inside the parentheses:

$$0.75(-280) = 0.35(-300) + 0.35(-300)$$

Next, perform the multiplications:

$$-210 = -105 - 105$$

Finally, simplify the right side of the equation:

$$-210 = -210$$

Hey, it checks out! Both sides of the equation are equal, which means our solution, x = -320, is correct. This step is about verifying that our answer is mathematically sound. It's about developing a critical approach to our work, ensuring that we've not only solved the equation but that our solution is valid and reliable. Always remember to check your solutions; it's a great habit that saves you from making mistakes.

The Importance of Verification

Checking your solution is a smart move, guys! It helps you catch any calculation errors and builds your confidence. When you know your answer is correct, you can move forward with confidence. Moreover, this approach to problem-solving is not only useful for math; it’s a great practice in many other areas of life. It’s like double-checking your work before submitting it. It gives you peace of mind and builds your reputation for accuracy. Checking solutions makes you a better problem solver and strengthens your grasp of the material.

The Verdict: How Many Solutions?

So, after all our hard work, we've found that the equation $0.75(x+40)=0.35(x+20)+0.35(x+20)$ has one solution. We found that solution to be x = -320, and we verified that this value satisfies the equation. This type of equation, where we get a single, distinct value for 'x', is a common characteristic of linear equations. It means there is one specific point where the left and right sides of the equation balance each other perfectly. This confirms that, for this specific equation, there's only one value of 'x' that makes the equation true. Knowing this helps to understand the behavior of the equation and its role in solving mathematical problems.

Recap and Key Takeaways

• We broke down the equation step by step, applying algebraic principles to isolate 'x'.
• We solved for x, finding a single value that satisfies the equation.
• We verified our answer by plugging the solution back into the original equation.
• We determined that the equation has only one solution.

This entire process is a foundational skill in mathematics. It emphasizes the importance of accuracy, step-by-step thinking, and validation of your results. By mastering these skills, you are well-prepared for more complex mathematical problems. Understanding how to solve equations is a fundamental skill that underpins many areas of mathematics and its applications in real life. Keep practicing and keep up the great work, everyone!