Solving The Algebraic Equation -36 + 50 + 15x = 5x - 10x + 25 + 32

July 29, 2025 by Scholario Team 67 views

Unraveling the Mathematical Equation: -36 + 50 + 15x = 5x - 10x + 25 + 32

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of algebra to dissect and solve the equation -36 + 50 + 15x = 5x - 10x + 25 + 32. This equation might look a bit intimidating at first glance, but don't worry, we'll break it down step by step, making it as clear as crystal. Our goal? To find the value of 'x' that makes this equation true. So, grab your calculators, sharpen your pencils, and let's get started on this mathematical adventure!

Understanding the Basics of Algebraic Equations

Before we jump into solving this specific equation, let's take a moment to refresh our understanding of the fundamental principles of algebraic equations. In essence, an algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions can involve numbers, variables (like our 'x'), and mathematical operations such as addition, subtraction, multiplication, and division. The magic of solving an equation lies in isolating the variable on one side of the equation to determine its value.

The key concept here is the principle of equality: whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain the balance. Think of it like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. This principle is our guiding star as we navigate the equation -36 + 50 + 15x = 5x - 10x + 25 + 32.

Another crucial aspect is combining like terms. Like terms are terms that contain the same variable raised to the same power (or are just constants). For example, in our equation, '15x', '5x', and '-10x' are like terms because they all contain 'x' raised to the power of 1. Similarly, '-36', '50', '25', and '32' are like terms because they are all constants. Combining like terms simplifies the equation, making it easier to solve. Remember, we can only add or subtract like terms; we can't combine 'x' terms with constant terms.

With these basics in mind, we're well-equipped to tackle our equation. We'll use the principle of equality and the concept of combining like terms to systematically isolate 'x' and find its value. So, let's roll up our sleeves and dive into the solution!

Step-by-Step Solution of the Equation

Now, let's get our hands dirty and solve the equation -36 + 50 + 15x = 5x - 10x + 25 + 32 step by step. Remember, the key is to maintain balance and simplify along the way.

Step 1: Simplify Both Sides by Combining Like Terms

The first thing we want to do is make both sides of the equation as simple as possible. This means combining the constant terms on each side and combining the 'x' terms on each side.

On the left side, we have '-36' and '50' as constants. Combining these, we get:

-36 + 50 = 14

So, the left side of the equation simplifies to:

14 + 15x

On the right side, we have '5x' and '-10x' as 'x' terms, and '25' and '32' as constants. Let's combine these:

5x - 10x = -5x

25 + 32 = 57

So, the right side of the equation simplifies to:

-5x + 57

Now, our equation looks much cleaner:

14 + 15x = -5x + 57

Step 2: Isolate the 'x' Terms on One Side

Our next goal is to get all the 'x' terms on one side of the equation and the constants on the other side. It doesn't matter which side we choose for the 'x' terms, but it's often easier to choose the side that will result in a positive coefficient for 'x'. In this case, we can add '5x' to both sides to eliminate the '-5x' term on the right side:

14 + 15x + 5x = -5x + 57 + 5x

This simplifies to:

14 + 20x = 57

Step 3: Isolate the Constant Terms on the Other Side

Now that we have all the 'x' terms on the left side, we want to get all the constant terms on the right side. To do this, we can subtract '14' from both sides:

14 + 20x - 14 = 57 - 14

This simplifies to:

20x = 43

Step 4: Solve for 'x'

We're almost there! Now we have '20x = 43'. To solve for 'x', we need to isolate 'x' by dividing both sides by '20':

20x / 20 = 43 / 20

This gives us:

x = 43 / 20

So, the solution to the equation is x = 43 / 20, which can also be written as x = 2.15 in decimal form.

Verification of the Solution

It's always a good idea to verify our solution to make sure we haven't made any mistakes along the way. To do this, we substitute our value of 'x' back into the original equation and see if both sides are equal.

Our original equation is:

-36 + 50 + 15x = 5x - 10x + 25 + 32

Substituting x = 43 / 20 into the equation, we get:

-36 + 50 + 15(43 / 20) = 5(43 / 20) - 10(43 / 20) + 25 + 32

Let's simplify both sides:

Left Side:

-36 + 50 + (15 * 43) / 20 = 14 + 645 / 20 = 14 + 32.25 = 46.25

Right Side:

(5 * 43) / 20 - (10 * 43) / 20 + 25 + 32 = 215 / 20 - 430 / 20 + 57 = 10.75 - 21.5 + 57 = 46.25

Since both sides equal 46.25, our solution x = 43 / 20 is correct! We've successfully verified our answer.

Common Mistakes to Avoid

When solving algebraic equations, it's easy to make mistakes if we're not careful. Here are some common pitfalls to watch out for:

Not applying the principle of equality correctly: Remember, whatever operation you perform on one side of the equation, you must perform the same operation on the other side. Failing to do so will throw off the balance and lead to an incorrect solution.
Incorrectly combining like terms: Only terms with the same variable raised to the same power (or constants) can be combined. For example, you can combine '3x' and '5x', but you can't combine '3x' and '5x²'. Also, be mindful of the signs (+ or -) when combining terms.
Forgetting to distribute: If you have a term multiplied by an expression in parentheses, you need to distribute the term to each term inside the parentheses. For example, if you have '2(x + 3)', you need to multiply '2' by both 'x' and '3', resulting in '2x + 6'.
Making arithmetic errors: Simple addition, subtraction, multiplication, or division errors can derail your solution. Double-check your calculations to ensure accuracy.
Not verifying the solution: It's always a good practice to substitute your solution back into the original equation to verify that it works. This can help you catch any mistakes you might have made along the way.

By being aware of these common mistakes, you can significantly improve your accuracy and confidence in solving algebraic equations.

Real-World Applications of Algebraic Equations

You might be wondering, "Why are we learning this? Where will I ever use algebraic equations in the real world?" Well, the truth is, algebraic equations are incredibly versatile and have countless applications in various fields. Let's explore a few examples:

Science and Engineering: Algebraic equations are the backbone of many scientific and engineering calculations. They're used to model physical phenomena, design structures, analyze circuits, and much more. For instance, engineers use equations to calculate the stress and strain on a bridge, while physicists use them to describe the motion of objects.
Finance and Economics: From calculating interest rates on loans to predicting market trends, algebraic equations play a crucial role in finance and economics. Economists use them to model supply and demand, while financial analysts use them to assess investment risks and returns.
Computer Science: Computer programs rely heavily on algebraic equations to perform calculations and make decisions. From simple arithmetic operations to complex algorithms, equations are the language of computers.
Everyday Life: Believe it or not, you use algebraic equations in your daily life more often than you might realize. For example, when you're calculating the total cost of items at the store, determining how much time it will take to drive to a destination, or even adjusting a recipe, you're implicitly using algebraic principles.
Problem Solving: Beyond specific applications, learning to solve algebraic equations develops valuable problem-solving skills that can be applied in any field. The ability to break down a problem into smaller parts, identify patterns, and use logical reasoning is essential for success in many areas of life.

So, the next time you're solving an algebraic equation, remember that you're not just learning a math skill – you're developing a powerful tool that can be used to understand and solve problems in the real world.

Conclusion: Mastering the Art of Equation Solving

We've reached the end of our journey through the equation -36 + 50 + 15x = 5x - 10x + 25 + 32. We've broken it down, solved for 'x', verified our solution, and even explored some real-world applications of algebraic equations. Hopefully, you now feel more confident and comfortable tackling similar challenges.

Remember, the key to mastering equation solving is practice. The more you work with equations, the better you'll become at recognizing patterns, applying the correct steps, and avoiding common mistakes. Don't be afraid to make mistakes – they're a natural part of the learning process. Just be sure to learn from them and keep practicing.

Algebraic equations might seem abstract at times, but they're a powerful tool for understanding the world around us. They help us model relationships, solve problems, and make informed decisions. So, keep honing your skills, and you'll be amazed at what you can achieve!

If you have any more questions or want to explore other mathematical concepts, feel free to ask. Keep learning, keep exploring, and keep having fun with math!