Solving Equations: A Step-by-Step Guide

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Solving Equations: A Step-by-Step Guide

Hey guys! Let's dive into how to solve equations like -5(3x - 6) = 2(-3x - 5). This might look a bit intimidating at first, but trust me, it's totally manageable if you break it down into steps. We're going to go through this super carefully, so you can totally nail these problems. Understanding how to solve equations is a fundamental skill in math, so let's get started. We'll start with the distributive property, then combine like terms, isolate the variable, and finally solve for x. This step-by-step approach will help you conquer any equation, so pay close attention. Ready? Let's get this show on the road!

Step 1: Distribute and Conquer

The first step in solving our equation -5(3x - 6) = 2(-3x - 5) is to use the distributive property. This means we need to multiply the numbers outside the parentheses by each term inside the parentheses. Think of it like this: the number outside has to 'visit' everyone inside. So, for the left side of the equation, we multiply -5 by both 3x and -6. For the right side, we multiply 2 by both -3x and -5. Let's break it down:

  • Left Side: -5 * (3x) = -15x and -5 * (-6) = 30. So, the left side becomes -15x + 30.
  • Right Side: 2 * (-3x) = -6x and 2 * (-5) = -10. So, the right side becomes -6x - 10.

Now, our equation looks like this: -15x + 30 = -6x - 10. See? Much less scary already! This initial distribution is super important because it clears the parentheses, making it easier to combine like terms. This process is key because it simplifies the equation into a more manageable form. Think of it as preparing all your ingredients before you start cooking. We're getting the equation ready to solve for x. Remember, the distributive property is used to eliminate parentheses in the expression, ensuring we can simplify and solve the problem accurately. This initial distribution is key to simplifying the expression and ensuring accurate calculations. Once we have distributed properly, the next steps will be much clearer.

Step 2: Combine the Like Terms

Now that we have distributed, our next move is to get all the 'x' terms on one side of the equation and all the constant numbers on the other side. This is like sorting your socks – all the matching ones together! To do this, we need to move terms across the equals sign. When you move a term across the equals sign, you have to change its sign. This is super important, so don't forget it.

Let's start by moving the -6x from the right side to the left side. To do this, we add 6x to both sides of the equation. This cancels out the -6x on the right side and adds 6x to the left side. Our equation now looks like this: -15x + 6x + 30 = -10. Cool, right?

Next, we need to move the constant term (the number without an 'x') from the left side to the right side. In our case, that's the 30. We subtract 30 from both sides of the equation. This cancels out the +30 on the left side and subtracts 30 from the right side. Our equation now looks like this: -15x + 6x = -10 - 30.

Now we can combine the like terms: -15x + 6x equals -9x, and -10 - 30 equals -40. So, our equation simplifies to -9x = -40. Combining like terms is all about simplifying the equation to find the value of x. This simplification will lead us to the final solution by isolating the variable and performing the appropriate operations. Remember, the goal is always to isolate the variable so you can find its value. So, we're making progress. Keep it up! This step is key in bringing us closer to finding the value of x.

Step 3: Isolate the Variable

We're getting close to the finish line! Now that we've combined like terms, we have the equation -9x = -40. Our next goal is to isolate 'x'. This means we want to get 'x' all by itself on one side of the equation. To do this, we need to get rid of the -9 that's being multiplied by x.

The way to do this is to perform the opposite operation. Since -9 is multiplying x, we do the opposite: divide both sides of the equation by -9. Remember, whatever you do to one side of the equation, you have to do to the other side to keep things balanced. So, we divide both sides by -9:

  • (-9x) / -9 = x
  • (-40) / -9 = 40/9

This leaves us with x = 40/9. We've done it! We've isolated the variable and found the value of x. Isolating the variable is like the grand finale – this step directly leads to the final solution. At this point, you're almost done. You can see we're just about to find the final value of x. The key is to get x by itself. Remember, keep both sides balanced by performing the same operations.

Step 4: Solve for x (Find the Solution)

We've already done the hard part! The final step is to solve for x, which means finding the actual value of x. In our equation, we ended up with x = 40/9. This is our solution. You can express this as a fraction (40/9) or, if you prefer, as a decimal. To get the decimal, just divide 40 by 9, which gives you approximately 4.44. So, x is approximately 4.44. Congratulations, you've solved the equation! The value of x represents the number that makes the original equation true. Substituting this value back into the original equation will confirm its validity. We have isolated the variable, and the answer is right in front of us. Make sure to simplify the answer as much as possible, as we did in the above steps. This step makes sure you have a clear and accurate final answer. You should make sure you check your answer to verify it works.

Step 5: Verify the Answer

It's always a good practice to verify your answer by substituting the value of x back into the original equation. This is like a double-check to make sure you didn't make any mistakes. Let's do it with our original equation: -5(3x - 6) = 2(-3x - 5). Remember, we found that x = 40/9. Let's substitute that into the equation:

  • -5(3 * (40/9) - 6) = 2(-3 * (40/9) - 5)

Now, let's simplify:

  • -5((120/9) - 6) = 2((-120/9) - 5)
  • -5((120/9) - (54/9)) = 2((-120/9) - (45/9))
  • -5(66/9) = 2(-165/9)
  • -330/9 = -330/9

Since both sides of the equation are equal, our solution is correct! Checking the answer guarantees the validity of the solution and offers confidence. This process ensures the accuracy of our calculations, leaving no space for error. Remember that in some cases, the values will be approximate if you use decimals. The verification step is crucial for reinforcing your understanding and ensuring accuracy in problem-solving. It's like a final quality check, confirming that our solution is correct.

Summary

  • Distribution: Use the distributive property to eliminate parentheses.
  • Combine Like Terms: Group all the 'x' terms on one side and the constants on the other.
  • Isolate the Variable: Get 'x' by itself by performing the opposite operation.
  • Solve for x: Calculate the value of x.
  • Verify: Substitute the value of x back into the original equation to check your answer.

Well done! You have successfully navigated the process of solving equations. Keep practicing, and you'll become a pro in no time! Remember to always follow the steps, and you'll be well on your way to math mastery. You've now gained valuable skills that you can apply to various mathematical problems. By repeatedly practicing the steps, you'll feel more confident about your ability to solve all of them.