Solving 8x - 5 = 19 Step-by-Step A Comprehensive Guide

Hey guys! Ever get stuck trying to solve a simple equation? Don't worry, we've all been there! Today, we're going to break down a classic linear equation and walk through the correct way to solve it. We'll use the example 8x - 5 = 19 as our guide. So, buckle up and let's dive in!

Understanding Linear Equations

Before we jump into the solution, let's quickly recap what a linear equation actually is. Basically, it's an equation where the highest power of the variable (in our case, 'x') is 1. These equations represent a straight line when graphed, hence the name "linear." They are the building blocks of algebra and pop up everywhere in math and science. To properly solve these kinds of equations it's fundamental to use reverse operations. Operations such as PEMDAS/BODMAS are fundamental to solving this equation. The operations have to be made reverse to isolate x on one side of the equation and solve for the unknown variable.

The core idea behind solving any equation is to isolate the variable. This means getting the 'x' all by itself on one side of the equals sign. To do this, we use inverse operations. Think of it like undoing a series of actions. If someone added 5, we subtract 5. If someone multiplied by 8, we divide by 8. The golden rule is: whatever you do to one side of the equation, you MUST do to the other side to keep things balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level. Now that we understand the concept let's see how to apply to a practical example.

Step-by-Step Solution of 8x - 5 = 19

Let's solve the equation 8x - 5 = 19 step-by-step, making sure to explain each move. So our goal, as we mentioned, is to get x on its own side of the equation.

Step 1: Isolate the term with 'x'

The first thing we need to do is get rid of the '-5' on the left side. Since it's being subtracted, we'll do the opposite – we'll add 5 to both sides of the equation. This is our first application of the golden rule. By adding 5 on both sides of the equation will isolate 8x from the rest of the expression on the left side and start solving the equation. So:

8x - 5 = 19
8x - 5 + 5 = 19 + 5

This simplifies to:

8x = 24

See? The '-5' is gone from the left side, and we've added 5 to the right side to keep everything balanced. It's like magic, but it's just math! Now we are one step closer to solving the equation by having 8x on one side of the equation.

Step 2: Isolate 'x'

Now we have 8x = 24. The 'x' is being multiplied by 8. To undo this multiplication, we'll do the opposite: we'll divide both sides of the equation by 8. This is the second inverse operation that we apply in this equation:

8x = 24
(8x) / 8 = 24 / 8

This simplifies to:

x = 3

Boom! We've done it! We've isolated 'x' and found our solution. x = 3 is the solution to our equation. It means that if we substitute 3 for x in the original equation, we will make it true. Let's check that!

Step 3: Check your solution

It's always a good idea to check your answer to make sure you didn't make any silly mistakes. To do this, we substitute our solution (x = 3) back into the original equation 8x - 5 = 19:

8 * (3) - 5 = 19
24 - 5 = 19
19 = 19

It works! The left side equals the right side, so our solution x = 3 is correct. This step is essential to verify that the solution obtained is correct, this way we avoid mistakes in our calculations and guarantee the solution is the right one. Remember to always verify your solutions.

Common Mistakes to Avoid

Solving linear equations is pretty straightforward once you get the hang of it, but there are a few common pitfalls to watch out for:

• Forgetting the Golden Rule: The most common mistake is not doing the same operation to both sides of the equation. This throws off the balance and leads to a wrong answer.
• Incorrect Order of Operations: Remember to undo operations in the reverse order of operations (PEMDAS/BODMAS). So, we deal with addition and subtraction before multiplication and division. Applying the correct order of operations is critical to ensure the correct answer is obtained.
• Sign Errors: Be extra careful with negative signs. A small mistake with a sign can completely change the solution.

Why is This Important?

You might be thinking, "Okay, I can solve this equation… but why does it even matter?" Well, linear equations are the foundation for a LOT of things! They show up in physics, engineering, economics, computer science… basically any field that uses math. Understanding how to solve them is a crucial skill for problem-solving in general. Linear equations are used to modeling real world situations so it's a skill that will help you for your life.

Let's Recap

To sum it all up, solving linear equations is about isolating the variable using inverse operations while maintaining balance. We walked through the steps with the example 8x - 5 = 19, and we saw how important it is to check our solution. Keep practicing, and you'll be solving these equations like a pro in no time! You can always break it down step by step, identify the operations and apply its inverse to solve the equation. Practice makes perfect.

Practice Problems

Want to test your skills? Try solving these equations:

1. 3x + 7 = 22
2. 5x - 9 = 11
3. 2x + 1 = 9

Share your solutions in the comments below! We can discuss your approach and the result.

Conclusion

So, there you have it! Solving linear equations doesn't have to be scary. By understanding the basic principles and practicing regularly, you can master this essential skill. Remember the golden rule, watch out for common mistakes, and always check your answers. You've got this! Keep up the great work, and happy problem-solving! We hope this helps you to understand how to solve equations and the logic behind the operations. Always remember to practice, it is the best way to improve your skills and apply this concepts to other problems.

Which of the following options correctly demonstrates the solution steps for the equation $8x - 5 = 19$? Let's dissect this problem step-by-step to identify the correct approach and solution. Solving linear equations is a fundamental skill in algebra, and understanding the process thoroughly is crucial for tackling more complex mathematical problems. So, let's break it down, guys!

The Importance of Showing Solution Steps

In mathematics, especially in algebra, it's not just about arriving at the correct answer; it's equally important to understand and demonstrate the process of how you got there. Showing your solution steps allows you to:

• Track Your Work: By writing down each step, you can easily review your work and identify any potential errors.
• Communicate Your Reasoning: Clear steps demonstrate your understanding of the underlying mathematical principles to others.
• Build a Strong Foundation: Mastering the step-by-step process reinforces your understanding and prepares you for more advanced topics.

Analyzing the Equation: 8x - 5 = 19

Our task is to solve the linear equation $8x - 5 = 19$. This means we need to isolate the variable 'x' on one side of the equation. Remember, the key principle here is to perform the same operations on both sides of the equation to maintain balance. It’s like a scale – if you add or subtract something from one side, you must do the same on the other side to keep it balanced.

Correct Solution Steps

Let's walk through the correct solution steps, explaining the rationale behind each step:

Step 1: Isolate the Term with 'x'

The first goal is to isolate the term containing 'x', which is '8x'. To do this, we need to eliminate the '-5' on the left side. The inverse operation of subtraction is addition, so we will add 5 to both sides of the equation:

8x - 5 = 19
8x - 5 + 5 = 19 + 5

This simplifies to:

8x = 24

By adding 5 to both sides, we have successfully isolated the term '8x'. We've moved one step closer to solving for 'x'. Now we have a simpler equation to deal with.

Step 2: Isolate 'x'

Now we have the equation $8x = 24$. The variable 'x' is being multiplied by 8. To isolate 'x', we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by 8:

8x = 24
(8x) / 8 = 24 / 8

This simplifies to:

x = 3

Voilà! We have found the solution: $x = 3$. We have successfully isolated 'x' and determined its value. But, as we've said before, it's crucial to check our solution to ensure accuracy.

Step 3: Verify the Solution

To verify our solution, we substitute $x = 3$ back into the original equation:

8x - 5 = 19
8(3) - 5 = 19
24 - 5 = 19
19 = 19

The left side of the equation equals the right side, confirming that our solution $x = 3$ is correct. This verification step is a critical part of the problem-solving process.

Incorrect Solution Attempts: Spotting the Errors

It's helpful to understand not just the correct solution but also common errors that students might make. Let's consider a potential incorrect solution attempt and analyze why it's wrong.

Example of an Incorrect Step:

8x - 5 = 19
8x - 5 + 5 = 19 - 5  // Incorrect

The error here is that 5 is added to the left side, but it is subtracted from the right side. This violates the fundamental principle of maintaining balance in the equation. Whatever operation is performed on one side must be performed on the other side.

Why Understanding the Process Matters

Understanding the correct steps and the rationale behind them is crucial for building a strong foundation in algebra. It's not enough to just memorize the steps; you need to understand why they work. This understanding will allow you to:

• Solve a Variety of Problems: You'll be able to adapt your approach to different types of linear equations.
• Avoid Common Mistakes: By understanding the principles, you'll be less likely to make errors.
• Tackle More Complex Equations: Linear equations are the building blocks for more advanced algebraic concepts, so a strong foundation here is essential.

Practical Applications of Linear Equations

Linear equations aren't just abstract mathematical concepts; they have numerous real-world applications. They are used to model relationships between quantities, solve problems in physics, engineering, economics, and many other fields. For example:

• Calculating Costs: Determining the total cost of items based on a fixed price per item and a fixed service fee.
• Distance and Speed Problems: Calculating the time it takes to travel a certain distance at a constant speed.
• Mixture Problems: Finding the amount of different solutions to mix to achieve a desired concentration.

Conclusion

In conclusion, the correct solution steps for $8x - 5 = 19$ involve adding 5 to both sides, then dividing both sides by 8, resulting in the solution $x = 3$. It's essential to understand the process behind solving equations, not just the final answer. Showing your steps, checking your solution, and understanding common errors are all crucial aspects of mastering algebra. So, keep practicing, keep asking questions, and you'll become a pro at solving linear equations! Understanding how to solve equations is a powerful skill that will help you in many areas of life. Remember the importance of verification to guarantee that the solution is the right one. Keep your equations balanced and you'll be ok! Good luck!