Hey guys! Let's dive into this math problem together. We're going to break down how Heidi solved the equation . We'll look at each step she took and figure out exactly what she did to get there. Math can seem tricky sometimes, but when we take it one step at a time, it becomes way easier to understand. So, let's put on our detective hats and get started!
The Original Equation
Before we get into the step-by-step breakdown, let's take a quick look at the equation Heidi was working with:
This might look a little intimidating at first glance, but don't worry! We're going to unpack it. The goal here is to figure out the value of 'x' that makes this equation true. To do that, Heidi used a series of algebraic steps to isolate 'x' on one side of the equation. We're going to walk through those steps and see why each one is valid.
Step 1 Unveiled Applying the Distributive Property
Okay, so Heidi's first move was:
So, what exactly happened here? This is a classic example of using the distributive property. Remember that this property tells us how to multiply a single term by a group of terms inside parentheses. In this case, Heidi distributed the '3' across the and the '5' across the .
Let's break it down even further:
- Distributing the 3: She multiplied the 3 by both the 'x' and the '4' inside the first set of parentheses. 3 times x is 3x, and 3 times 4 is 12. That's how we get the on the left side.
- Distributing the 5: Similarly, she multiplied the 5 by both the 'x' and the '-4' inside the second set of parentheses. 5 times x is 5x, and 5 times -4 is -20. That's where the on the right side comes from.
Essentially, the distributive property lets us get rid of the parentheses, which is a crucial step in solving for 'x'. It transforms the equation into a form that's easier to manipulate. By applying this property, we maintain the balance of the equation, ensuring that both sides remain equal.
This step is super important because it sets the stage for simplifying the equation further. If Heidi hadn't used the distributive property correctly, the rest of the solution would be off. So, give yourself a pat on the back if you spotted this one! You're thinking like a mathematician.
Step 2 Simplifying the Equation Combining Like Terms
Next up, Heidi went from:
What changed? In this step, Heidi simplified both sides of the equation by combining like terms. This means she grouped together the constant terms on each side to make the equation cleaner and easier to work with.
Let's see how it works:
- Left Side: On the left side of the equation (), Heidi had two constant terms: 12 and 2. She simply added these together (12 + 2 = 14) to get the simplified expression .
- Right Side: The right side of the equation () also had constant terms: 2 and -20. Adding these together (2 - 20 = -18) resulted in the simplified expression .
Combining like terms is a fundamental technique in algebra. It helps to reduce the complexity of an equation by gathering similar terms. This makes the equation easier to read and less prone to errors when we perform further operations. It's like tidying up a messy room before you start organizing things – it just makes the whole process smoother!
By combining like terms, Heidi has taken another step towards isolating 'x'. The equation is now more streamlined, and we're closer to finding the solution. Notice how each step builds upon the previous one, gradually simplifying the equation until we can finally solve for 'x'.
Step 3 Isolating the Variable Moving Terms Across the Equal Sign
Now, let's look at step 3:
How did Heidi get there? In this step, Heidi strategically moved terms around to isolate the variable 'x' on one side of the equation. This involves using the properties of equality to maintain the balance of the equation while shifting terms from one side to the other.
Here's the breakdown:
- Subtracting 3x from both sides: To get the 'x' terms on the right side, Heidi subtracted from both sides of the equation. This is a crucial step because it keeps the equation balanced. Whatever you do to one side, you must do to the other. Subtracting from the left side () leaves us with just 14. Subtracting from the right side () gives us (since ).
The key idea here is the addition property of equality. This property states that you can add (or subtract) the same value from both sides of an equation without changing its solution. By subtracting from both sides, Heidi effectively moved the 'x' term to the right side of the equation while maintaining the equality.
Isolating the variable is a core strategy in solving equations. It's like separating the pieces of a puzzle so you can see them clearly and put them together. By getting all the 'x' terms on one side, Heidi is making it easier to eventually solve for 'x'.
Step 4 Continuing to Isolate Adding a Constant to Both Sides
Let's examine step 4:
What was Heidi's reasoning here? In this step, Heidi continued to isolate 'x' by getting rid of the constant term (-18) on the right side of the equation. She did this by adding 18 to both sides.
Let's see why this works:
- Adding 18 to both sides: Heidi added 18 to both the left side (14) and the right side () of the equation. On the left side, 14 + 18 equals 32. On the right side, adding 18 cancels out the -18, leaving just (since ).
Again, we're using the addition property of equality here. By adding the same value to both sides, we maintain the balance of the equation. Adding 18 was the perfect move because it eliminated the constant term on the right side, bringing us closer to having 'x' all by itself.
This step demonstrates the importance of inverse operations. Addition and subtraction are inverse operations, meaning they undo each other. By adding 18, Heidi effectively "undid" the subtraction of 18, isolating the term with 'x'.
At this point, the equation is looking much simpler! We're just one step away from finding the value of 'x'. Heidi has done a great job of strategically manipulating the equation to get us here.
Step 5 The Grand Finale Solving for x by Dividing
Finally, we arrive at step 5:
How did Heidi solve for x? This is the final step where Heidi isolates 'x' completely by dividing both sides of the equation by 2.
Here's the logic:
- Dividing both sides by 2: Heidi divided both the left side (32) and the right side () by 2. On the left side, 32 divided by 2 is 16. On the right side, divided by 2 is simply 'x'.
This step uses the division property of equality, which states that you can divide both sides of an equation by the same non-zero value without changing its solution. Dividing by 2 was the key move because it undid the multiplication of 'x' by 2, leaving 'x' all by itself.
And there you have it! Heidi has successfully solved the equation. The solution is . This means that if you substitute 16 for 'x' in the original equation, both sides will be equal.
Conclusion Heidi's Algebraic Triumph
So, let's recap what we've learned from Heidi's equation-solving journey. By carefully applying algebraic properties step-by-step, she transformed a seemingly complex equation into a simple solution. She used the distributive property, combined like terms, and strategically moved terms around using the addition, subtraction, and division properties of equality.
Each step was crucial in simplifying the equation and isolating the variable 'x'. By understanding the reasoning behind each step, we can tackle similar problems with confidence. Math isn't just about getting the right answer; it's about understanding the process and the logic involved.
Great job, Heidi, and great job to all of you for following along! Keep practicing, and you'll become equation-solving pros in no time. Remember, math is like a puzzle, and each step is a piece that fits together to reveal the solution.