Solving 9x + 7x - 4x - 5 = 19 Step By Step Guide

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Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a jumbled mess of numbers and variables? Don't worry, we've all been there. Today, we're going to break down the equation 9x + 7x - 4x - 5 = 19 step by step, so you can conquer it with confidence. This isn't just about getting the answer; it's about understanding the process and building your problem-solving skills. So, let's dive in and make math a little less intimidating and a lot more fun!

Understanding the Basics of Algebraic Equations

Before we jump into solving our specific equation, let's quickly recap some fundamental concepts. Think of an algebraic equation as a balanced scale. The equals sign (=) is the fulcrum, and both sides of the equation must always remain balanced. Our goal is to isolate the variable, in this case, 'x', on one side of the equation to find its value. To do this, we use inverse operations – operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. Keeping this balance analogy in mind will help you visualize the steps we take to solve the equation.

In our equation, 9x + 7x - 4x - 5 = 19, we have several terms involving 'x' and a constant term (-5) on the left side, and a constant (19) on the right side. Our strategy will be to first combine the 'x' terms, then isolate the variable term, and finally solve for 'x'. Remember, each step we take must maintain the balance of the equation. This means performing the same operation on both sides. It's like adding or removing the same weight from both sides of the scale to keep it level.

Furthermore, it’s important to understand the concept of like terms. Like terms are terms that have the same variable raised to the same power. In our equation, 9x, 7x, and -4x are like terms because they all have 'x' raised to the power of 1. We can combine like terms by simply adding or subtracting their coefficients (the numbers in front of the variables). This simplification is a crucial step in solving algebraic equations efficiently. Neglecting to combine like terms can lead to unnecessary complexity and increase the chances of making errors.

Step-by-Step Solution: 9x + 7x - 4x - 5 = 19

Alright, let's get down to business and solve this equation. We'll break it down into manageable steps, so you can follow along easily. Remember, the key is to maintain balance and apply inverse operations strategically.

Step 1: Combine Like Terms

The first thing we want to do is simplify the left side of the equation by combining the 'x' terms. We have 9x, 7x, and -4x. To combine them, we simply add and subtract their coefficients:

9x + 7x - 4x = (9 + 7 - 4)x = 12x

So, our equation now looks like this:

12x - 5 = 19

See how much simpler that is? Combining like terms helps to streamline the equation and makes the subsequent steps easier to manage. This is a crucial technique for tackling more complex algebraic problems.

Step 2: Isolate the Variable Term

Now, we want to isolate the term with 'x' (12x) on one side of the equation. To do this, we need to get rid of the constant term (-5) on the left side. Remember, we use inverse operations to do this. The inverse operation of subtraction is addition, so we'll add 5 to both sides of the equation:

12x - 5 + 5 = 19 + 5

This simplifies to:

12x = 24

Notice how adding 5 to both sides maintains the balance of the equation. We've successfully isolated the variable term, bringing us one step closer to solving for 'x'.

Step 3: Solve for x

The final step is to solve for 'x'. Currently, 'x' is being multiplied by 12. The inverse operation of multiplication is division, so we'll divide both sides of the equation by 12:

12x / 12 = 24 / 12

This gives us:

x = 2

And there you have it! We've successfully solved the equation. The value of 'x' that satisfies the equation 9x + 7x - 4x - 5 = 19 is 2.

Step 4: Verification

It is very important to verify the answer, the steps are as follows

Plug x = 2 back into the original equation:

9(2)+7(2)−4(2)−5=199(2) + 7(2) - 4(2) - 5 = 19

18+14−8−5=1918 + 14 - 8 - 5 = 19

32−13=1932 - 13 = 19

19=1919 = 19

The left and right sides of the equation are equal, so the answer is correct.

Common Mistakes and How to Avoid Them

Solving equations can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Forgetting to Distribute: If you have a term multiplied by an expression in parentheses, make sure to distribute it to every term inside the parentheses. For example, if you had 2(x + 3), you need to multiply both 'x' and '3' by 2.
  • Incorrectly Combining Like Terms: Only combine terms that have the same variable raised to the same power. For instance, you can combine 3x and 5x, but you can't combine 3x and 5x². Pay close attention to the signs (+ or -) when combining terms.
  • Not Applying Operations to Both Sides: Remember, the key to solving equations is maintaining balance. Any operation you perform on one side of the equation must also be performed on the other side. This ensures that the equation remains equal.
  • Sign Errors: Be extra careful with negative signs. A small sign error can throw off your entire solution. Double-check your work, especially when dealing with subtraction and division.
  • Skipping Steps: While it might be tempting to rush through the process, skipping steps can increase the likelihood of making mistakes. Write out each step clearly, especially when you're first learning. This will help you keep track of your work and identify any errors.

To avoid these mistakes, practice regularly, show your work, and double-check your answers. If you're unsure about a step, don't hesitate to ask for help or review the concepts involved. With consistent effort, you'll become more confident and accurate in your equation-solving abilities.

Practice Problems to Sharpen Your Skills

Now that we've solved our example equation and discussed common mistakes, it's time to put your knowledge to the test. Practice is key to mastering any mathematical skill. Here are a few similar equations for you to try:

  1. 5x + 3x - 2x + 7 = 21
  2. 10x - 4x + 6 - 2 = 16
  3. 3x + 8x - 5x - 10 = 20

Work through these problems step by step, applying the techniques we've discussed. Remember to combine like terms, isolate the variable term, and use inverse operations to solve for 'x'. Don't forget to check your answers by plugging them back into the original equations.

If you get stuck, review the steps we took to solve the example equation, or revisit the section on common mistakes. You can also find plenty of resources online and in textbooks to help you practice. The more you practice, the more comfortable and confident you'll become with solving algebraic equations.

Conclusion: Mastering Equations with Confidence

Solving equations is a fundamental skill in mathematics, and it's one that you'll use in many different contexts. By understanding the basic principles, practicing regularly, and avoiding common mistakes, you can master this skill and build a strong foundation for more advanced math topics. Remember, it's not just about getting the right answer; it's about understanding the process and developing your problem-solving abilities. So, keep practicing, stay curious, and enjoy the challenge of solving equations!

We've covered a lot in this guide, from understanding the basics of algebraic equations to solving the specific equation 9x + 7x - 4x - 5 = 19. We've also discussed common mistakes to avoid and provided practice problems to help you sharpen your skills. With dedication and effort, you can conquer any equation that comes your way. So, go forth and solve, and remember to have fun along the way!