Solving 2x + 5 = 19 Matching Steps With Justifications

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Hey guys! Let's dive into the exciting world of solving equations. It's like a puzzle where we need to find the hidden value of 'x'. But, it's not just about finding the answer; it's also about understanding why our steps are correct. Think of it as building a logical argument, where each step needs a solid justification. We're going to break down the equation $2x + 5 = 19$, matching each step with its reason. Get ready to become equation-solving pros!

The Foundation: Understanding Properties of Equality

Before we jump into the equation, let's lay the groundwork. The properties of equality are like the rules of the game. They tell us what we can legally do to an equation without changing its solution. Imagine an equation as a balanced scale; if you do something to one side, you must do the same to the other to keep it balanced. These properties are the justifications for our steps.

  • Addition Property of Equality: If you add the same number to both sides of an equation, the equation remains true. So, if a = b, then a + c = b + c. This is crucial for undoing subtraction. For instance, if we have x - 3 = 7, we can add 3 to both sides to isolate x.
  • Subtraction Property of Equality: You guessed it! If you subtract the same number from both sides, the equation stays balanced. If a = b, then a - c = b - c. This is perfect for undoing addition. If we have x + 5 = 12, we can subtract 5 from both sides.
  • Multiplication Property of Equality: Multiplying both sides by the same non-zero number keeps things equal. If a = b, then a * c = b * c (where c isn't zero). This is how we undo division. Imagine having x / 2 = 4; we can multiply both sides by 2.
  • Division Property of Equality: Dividing both sides by the same non-zero number also maintains equality. If a = b, then a / c = b / c (again, c can't be zero). This is how we undo multiplication. If we have 3x = 15, we can divide both sides by 3.

These properties are the why behind our equation-solving moves. Without them, we'd just be randomly changing the equation! Knowing these properties is like having a superpower in algebra. Now, let's apply these to our equation.

Step 1 Isolating the Variable Term

Our starting point is the equation $2x + 5 = 19$. The main goal in solving any equation is to isolate the variable – in this case, x. This means getting x all by itself on one side of the equation. To achieve this, we need to carefully peel away the layers of operations surrounding x, using the properties of equality as our tools.

Looking at our equation, the first thing we notice is the '+ 5' on the left side. This addition is preventing 2x from being isolated. The subtraction property of equality is the perfect tool to address this. By subtracting 5 from both sides of the equation, we can effectively undo the addition. Remember, whatever we do to one side, we must do to the other to maintain the balance of the equation. This is the golden rule of equation solving!

Subtracting 5 from both sides gives us: $2x + 5 - 5 = 19 - 5$. Simplifying this, we get $2x = 14$. Notice how the '+ 5' on the left side has disappeared, leaving us with just 2x. We've successfully isolated the term containing x. This step is crucial because it brings us one step closer to revealing the value of x. Think of it like carefully unwrapping a gift; we're slowly revealing what's inside.

The justification for this step is the subtraction property of equality. We are subtracting the same value (5) from both sides, ensuring the equation remains balanced and the solution unchanged. Understanding this justification is not just about memorizing a rule; it's about grasping the underlying logic of equation solving. Each step we take is a deliberate action with a clear reason, making the process both logical and understandable. By mastering this step, we've taken a significant stride towards solving the equation and solidifying our understanding of algebraic principles. This careful, step-by-step approach is what turns equation solving from a daunting task into a manageable and even enjoyable process.

Step 2: Isolating the Variable Itself

Now we've arrived at the equation $2x = 14$. We're close, but x isn't quite alone yet! The x is currently being multiplied by 2. To finally isolate x, we need to undo this multiplication. The property of equality that comes to our rescue here is the division property of equality. Just as subtraction undoes addition, division undoes multiplication.

The division property of equality states that if we divide both sides of an equation by the same non-zero number, the equation remains balanced. In our case, we need to divide both sides by 2. This will cancel out the multiplication by 2 on the left side, leaving us with just x. Remember, the key is to perform the same operation on both sides to maintain the equation's integrity. It's like ensuring both sides of a scale remain balanced; we can't just change one side without affecting the other.

Dividing both sides of $2x = 14$ by 2, we get: $\frac{2x}{2} = \frac{14}{2}$. Simplifying this, we find $x = 7$. Bingo! We've successfully isolated x and found its value. The multiplication by 2 has been undone by our division, leaving x all by itself. This step is the climax of our equation-solving journey; it's where the value of x is revealed.

The justification for this step is, as mentioned, the division property of equality. We divided both sides by 2, a valid operation that maintains the equation's balance. Understanding this justification reinforces the core principle of equation solving: performing the same operation on both sides to preserve equality. This step not only gives us the solution but also solidifies our understanding of the division property and its role in algebraic manipulation. With x isolated and its value determined, we can confidently say we've conquered this part of the equation-solving puzzle. This step is a testament to the power of applying the properties of equality strategically and logically.

Step 3: The Solution

So, after carefully applying the properties of equality, we've arrived at the solution: $x = 7$. But how do we know for sure that this is the correct answer? The best way to verify our solution is to substitute it back into the original equation. This is like checking our work to make sure everything adds up.

Our original equation was $2x + 5 = 19$. To check if $x = 7$ is the correct solution, we replace x with 7 in the equation: $2(7) + 5 = 19$. Now, we simplify the left side of the equation. First, we multiply: $2 * 7 = 14$. Then, we add: $14 + 5 = 19$. So, the left side simplifies to 19. Now we have $19 = 19$, which is a true statement! This confirms that $x = 7$ is indeed the correct solution.

Substitution is a powerful tool because it provides concrete evidence that our solution works. It's like the final piece of the puzzle that clicks into place, giving us a sense of completion and confidence in our answer. It also helps us catch any mistakes we might have made along the way. If the substitution doesn't result in a true statement, it means we need to go back and review our steps to find the error.

In the context of our problem, the justification for stating $x = 7$ as the solution is the successful verification through substitution. We've shown that when we replace x with 7 in the original equation, the equation holds true. This process not only validates our solution but also reinforces our understanding of what it means to solve an equation. It's not just about finding a number; it's about finding a number that makes the equation a true statement. This final step of verification is crucial for ensuring accuracy and solidifying our problem-solving skills. By substituting and confirming our solution, we complete the equation-solving process with confidence and precision.

Matching Steps and Justifications: The Grand Finale

Okay, let's put it all together and match each step with its correct justification. We've solved the equation and understand the reasoning behind each move. Now, it's time to connect the steps with the properties of equality we used.

Here's a recap of our steps:

  1. Original equation: $2x + 5 = 19$ (Given)
  2. Subtract 5 from both sides: $2x = 14$ (Subtraction Property of Equality)
  3. Divide both sides by 2: $x = 7$ (Division Property of Equality)

So, the matching looks like this:

  • x = 7$ is justified by the ***division property of equality***, as it's the result of dividing both sides of the previous equation by 2.

  • 2x = 14$ is the result of subtracting 5 from both sides of the original equation, so its justification is the ***subtraction property of equality***.

This matching exercise highlights the logical flow of equation solving. Each step is a direct consequence of the previous one, justified by a specific property of equality. It's like a chain reaction, where one action leads to the next, all guided by the principles of algebra. Understanding this connection between steps and justifications is key to mastering equation solving.

By correctly matching each step with its justification, we demonstrate a comprehensive understanding of the equation-solving process. It's not just about getting the right answer; it's about knowing why the answer is correct. This deeper understanding is what separates a good equation solver from a great one. We can confidently tackle any equation that comes our way. So, keep practicing, keep justifying your steps, and you'll become an equation-solving superstar!

Final Thoughts: The Power of Justification

Alright guys, we've journeyed through the equation $2x + 5 = 19$, dissected each step, and matched it with its justification. We've seen how the properties of equality act as the foundation for solving equations, providing the rules that allow us to manipulate equations while preserving their solutions. The ability to justify each step is what transforms equation solving from a mechanical process into a logical and understandable activity.

Remember, in mathematics, understanding why is just as important as understanding how. It's not enough to simply follow steps; we need to know the reasoning behind those steps. This is what allows us to apply our knowledge to new and challenging problems. When we understand the justifications, we're not just memorizing procedures; we're developing a deeper understanding of the underlying principles.

So, keep practicing, keep asking why, and keep building your equation-solving skills. The world of algebra is full of exciting challenges, and with a solid understanding of justifications, you'll be well-equipped to tackle them. Remember, every equation is a puzzle waiting to be solved, and every step is a logical move justified by the properties of equality. Go forth and conquer those equations!