Solving Equations: A Detailed Guide
Hey guys! Let's dive into solving some equations. We'll go through each problem step-by-step, making sure we understand everything clearly. Ready? Let's get started! This guide aims to provide a clear, comprehensive, and easy-to-understand walkthrough for solving the given algebraic equations. We'll break down each problem into manageable steps, explaining the rationale behind each operation to ensure a solid grasp of the concepts. This approach is designed to help you not only find the correct answers but also to understand why the steps work, building a strong foundation in algebra. The goal is to transform what might seem complex into something simple and intuitive. We'll focus on clarity, ensuring that even those new to algebra can follow along and gain confidence in their problem-solving abilities. So, let's roll up our sleeves and get into it! We'll start with the first equation and work our way through each one, ensuring you have a firm understanding of the methods used. Remember, the key to success in algebra, like many things, is practice, and by working through these problems together, we'll build your skills and confidence. This guide will provide a structured and supportive learning experience, making solving equations less intimidating and more enjoyable. Let's start with the first equation and break it down piece by piece to arrive at the solution. This detailed guide ensures you won't just solve equations but truly understand them.
Equation 1: 3x(x²-8) - 3x³ = 12
Alright, let's tackle the first equation: 3x(x²-8) - 3x³ = 12
. Our goal here is to isolate x. Here's how we'll do it:
- Expand the expression: First, we'll distribute the
3x
across the terms inside the parentheses. This means we multiply3x
byx²
and3x
by-8
. This gives us3x³ - 24x - 3x³ = 12
. - Simplify: Notice that we have
3x³
and-3x³
. These terms cancel each other out because they are opposites. This simplifies our equation to-24x = 12
. - Isolate x: To get
x
by itself, we need to divide both sides of the equation by-24
. So, we havex = 12 / -24
. - Solve for x: Finally, we perform the division to find the value of
x
.x = -0.5
or-1/2
.
So, the solution to the first equation is x = -0.5
. We started with expanding, simplified, isolated, and then solved for x
. Each step was designed to make the equation progressively simpler until we found our answer. Solving equations is like a puzzle, and each step brings us closer to the solution. The process is straightforward, and with practice, you'll become very comfortable with these types of problems. Remember, the key is to stay organized and perform each step carefully. Double-check your work, and you'll do great! We are simplifying the equation step by step, which helps to minimize errors and make the process more manageable. Make sure to keep the equation balanced by performing the same operations on both sides. This approach helps to build your confidence and understanding of solving algebraic equations. And there you have it – the complete solution to the first equation.
Equation 2: (x+8)(5x – 6) – 20 = 5x²
Let's move on to the second equation: (x+8)(5x – 6) – 20 = 5x²
. This one involves a bit more expansion, but don't worry, we'll break it down!
- Expand the parentheses: We'll start by expanding
(x+8)(5x – 6)
. This means multiplying each term in the first set of parentheses by each term in the second set. That gives us5x² - 6x + 40x - 48
. So, our equation now looks like5x² - 6x + 40x - 48 - 20 = 5x²
. - Simplify: Combine like terms. In this case, we can combine
-6x
and40x
to get34x
. Also, combine-48
and-20
to get-68
. Our equation becomes5x² + 34x - 68 = 5x²
. - Isolate x: Subtract
5x²
from both sides of the equation. This cancels out the5x²
terms, leaving us with34x - 68 = 0
. - Solve for x: Add 68 to both sides:
34x = 68
. Then, divide both sides by 34:x = 68 / 34
. Therefore,x = 2
.
So, the solution to the second equation is x = 2
. We expanded, simplified, and isolated x
using a logical progression. The expansion step is crucial, as it transforms the equation into a more manageable form. This process emphasizes the importance of carefully applying algebraic rules. Practice helps, but understanding why you're doing each step is just as important. With each step, we’re bringing the solution closer. Don't worry if it takes a bit of time to get comfortable; the more you practice, the easier it will become. The expansion and simplification steps are key to unlocking the solution. After that, isolating and solving for x becomes straightforward. We are always aiming to get to x, so that's the ultimate goal in all of these problems. Step-by-step, the equation unravels and the solution presents itself.
Equation 3: 18y³-2y(1 + 9y²) = 6,5
Alright, let's work on the third equation: 18y³-2y(1 + 9y²) = 6,5
. This one looks a little different, but the process is similar.
- Expand the expression: We need to distribute
-2y
across the terms inside the parentheses:-2y * 1
and-2y * 9y²
. This gives us-2y - 18y³
. So, our equation becomes18y³ - 2y - 18y³ = 6,5
. - Simplify: Notice the
18y³
and-18y³
cancel each other out. This simplifies our equation to-2y = 6,5
. - Isolate y: To get
y
by itself, divide both sides of the equation by-2
. Therefore,y = 6,5 / -2
. - Solve for y: Perform the division to find the value of
y
. This results iny = -3.25
or-13/4
.
Thus, the solution to the third equation is y = -3.25
. This shows how simplification can drastically change the equation, leading us to our solution. We used the same approach: expanding, simplifying, isolating, and solving. Each step is designed to bring us closer to the answer. The goal is to rearrange the equation to get y
alone on one side. Remember to be meticulous with the signs and the distribution of terms. We are systematically simplifying the equation to find the value of y
. The simplification steps are crucial here to see the cancellation of terms and focus on what remains. Take your time, and make sure each step is performed correctly. Always check the sign of the numbers and terms. We are well on our way to understanding how to solve these equations. We can see how crucial it is to get rid of the parentheses by using the distribution method. We are working our way through these equations one step at a time. This method is the foundation for further algebraic concepts.
Equation 4: 53-8y(1-3y) = 24y²
Let's finish up with the fourth equation: 53-8y(1-3y) = 24y²
. This one will wrap things up nicely!
- Expand the expression: First, we'll distribute
-8y
across the terms inside the parentheses. So,-8y * 1
and-8y * -3y
. This gives us-8y + 24y²
. Our equation now becomes53 - 8y + 24y² = 24y²
. - Simplify: Subtract
24y²
from both sides. This cancels out the24y²
terms, which simplifies the equation to53 - 8y = 0
. - Isolate y: Add
8y
to both sides:53 = 8y
. Now, divide both sides by 8:y = 53 / 8
. - Solve for y: Perform the division:
y = 6.625
or53/8
.
So, the solution to the fourth equation is y = 6.625
. We've applied the same principles throughout all four equations: expansion, simplification, isolation, and solving. You've now seen how to solve a variety of algebraic equations. Remember, the key is to stay organized and practice regularly. This structured approach helps you solve equations in a confident and systematic way. Keep practicing and applying these steps, and you'll get more comfortable and proficient with solving equations. You now have a solid foundation for tackling more complex algebraic problems. Each step is a building block that allows you to confidently solve more complex equations. From simplifying expressions to isolating the variable, you've learned to approach each problem systematically. You've got this, and you can solve many more equations. Keep practicing, and you'll become a pro in no time! Keep practicing, and you'll find that solving equations becomes a natural process.
Final Thoughts:
Congrats, guys! You've successfully worked through all the equations. Solving equations might seem intimidating at first, but by breaking them down into smaller, manageable steps, you've seen that it's totally doable. Remember to always double-check your work and to practice regularly. Each problem you solve builds your confidence and skills in algebra. Keep up the great work, and you'll find that solving equations becomes easier and more enjoyable over time! You are now equipped with the tools and strategies needed to solve a wide range of algebraic equations. Keep practicing and applying what you've learned. You're doing great, and with continued effort, you'll master algebra! The key is to stay consistent and persistent. Good luck, and keep up the amazing work! You now have a solid foundation for more advanced topics in mathematics! Keep up the practice, and success will surely follow! You've got all the tools you need to succeed! Keep practicing and stay positive, and you'll be amazed at what you can achieve. Keep learning and have fun with math!