Heidi's Equation: Step-by-Step Solution & Explanation

Hey math enthusiasts! Ever find yourself scratching your head over an equation? Let's break down a tricky one together! Today, we're diving into a problem solved by our friend Heidi, and we're going to dissect each step to understand the why behind it. So, grab your pencils and let's get started!

The Problem

Heidi tackled the equation:

3(x+4)+2=2+5(x-4)

And here's her solution:

3x + 12 + 2 = 2 + 5x - 20
3x + 14 = 5x - 18
14 = 2x - 18
32 = 2x
16 = x

But how did she get there? Let's break it down step-by-step and justify each move. We’ll explore the fundamental mathematical principles that allow us to manipulate equations while maintaining their balance and integrity. Understanding these principles is crucial for anyone looking to master algebra and problem-solving in mathematics.

Step 1: 3x + 12 + 2 = 2 + 5x - 20

Justification: The Distributive Property

The first step Heidi took was to expand the expressions on both sides of the equation. She did this by applying the distributive property. Remember, the distributive property states that a(b + c) = ab + ac. It's a fundamental concept in algebra that allows us to simplify expressions containing parentheses.

On the left side, she multiplied 3 by both x and 4 inside the parentheses: 3(x + 4) becomes 3 * x + 3 * 4, which simplifies to 3x + 12. On the right side, she did the same with 5(x - 4), which becomes 5 * x - 5 * 4, simplifying to 5x - 20. By applying the distributive property correctly, Heidi eliminated the parentheses, making the equation easier to work with. This is a common and essential technique in solving algebraic equations. Without this initial expansion, further simplification and isolation of the variable would be significantly more challenging.

This transformation is vital because it allows us to separate terms and eventually isolate the variable x. Imagine trying to solve the equation with the parentheses still in place—it would be like trying to untangle a knot without any slack! The distributive property gives us that necessary slack, allowing us to proceed with the solution. This step showcases the importance of recognizing and applying fundamental algebraic properties to simplify complex expressions and equations.

Step 2: 3x + 14 = 5x - 18

Justification: Combining Like Terms

In the second step, Heidi simplified both sides of the equation by combining like terms. Like terms are terms that have the same variable raised to the same power (or are just constants). Combining them makes the equation cleaner and easier to manage.

On the left side, she had 3x + 12 + 2. The constants 12 and 2 are like terms, so she added them together: 12 + 2 = 14. This simplifies the left side to 3x + 14. On the right side, the terms 2 and -20 are constants, so they can be combined. Adding these gives 2 + (-20) = -18, simplifying the right side to 5x - 18. Combining like terms is like sorting your socks – it makes everything neater and more organized! This step reduces the complexity of the equation, making it more approachable for further algebraic manipulation.

By grouping and combining like terms, we're essentially tidying up the equation, ensuring that we're only dealing with the essential components. It’s like trimming the excess foliage in a garden to allow the main plants to thrive. This step not only simplifies the appearance of the equation but also reduces the potential for errors in subsequent steps. Often overlooked, this seemingly small step plays a significant role in maintaining the accuracy and efficiency of the solution process. It highlights the importance of attention to detail and methodical simplification in algebraic problem-solving.

Step 3: 14 = 2x - 18

Justification: Subtraction Property of Equality

Here comes a crucial move! To isolate the terms with 'x' on one side, Heidi subtracted 3x from both sides of the equation. This is an application of the subtraction property of equality, which states that you can subtract the same value from both sides of an equation without changing its solution.

Starting with 3x + 14 = 5x - 18, she subtracted 3x from both sides. This cancels out the 3x on the left side, leaving just 14. On the right side, 5x - 3x simplifies to 2x. The equation now becomes 14 = 2x - 18. This subtraction strategically moves the variable term to one side of the equation, bringing us closer to solving for x. The subtraction property of equality ensures that the equation remains balanced, which is fundamental to finding the correct solution.

This maneuver is a key step in the algebraic dance of equation solving. It’s like strategically positioning pieces in a puzzle to reveal the final picture. By subtracting 3x from both sides, Heidi effectively grouped the x terms on the right, paving the way for isolating x. This step demonstrates the power of applying inverse operations to manipulate equations while maintaining balance. Understanding and applying such properties is essential for confidently navigating the world of algebra. It’s not just about getting the right answer; it’s about understanding the logical steps that lead to it.

Step 4: 32 = 2x

Justification: Addition Property of Equality

Now, let's get rid of that -18! Heidi added 18 to both sides of the equation. This is another application of the addition property of equality, which, like the subtraction property, ensures that adding the same value to both sides maintains the equation's balance.

Starting with 14 = 2x - 18, adding 18 to both sides cancels out the -18 on the right, leaving 2x. On the left side, 14 + 18 equals 32. So, the equation becomes 32 = 2x. This addition isolates the term with x on one side of the equation, bringing us closer to finding the value of x. The addition property of equality is a fundamental tool in algebra, allowing us to manipulate equations while preserving their solutions.

Think of this step as filling in the missing piece of the puzzle. By adding 18 to both sides, Heidi balanced the equation and brought us closer to isolating the variable. It's a clear demonstration of how inverse operations can be strategically used to simplify and solve equations. This step highlights the importance of recognizing and applying the properties of equality, which are cornerstones of algebraic manipulation. It’s about making the equation speak to us, revealing the value of x through logical and balanced transformations.

Step 5: 16 = x

Justification: Division Property of Equality

Almost there! To finally isolate x, Heidi divided both sides of the equation by 2. This is an application of the division property of equality, the last in our arsenal of equality properties. It states that dividing both sides of an equation by the same non-zero value preserves the solution.

Starting with 32 = 2x, dividing both sides by 2 isolates x on the right side: 2x / 2 = x. On the left side, 32 / 2 equals 16. Therefore, the equation simplifies to 16 = x. And there we have it – the solution! This division is the final step in unraveling the equation, revealing the value of x. The division property of equality ensures that this final step maintains the equation's balance and leads us to the correct solution.

This final move is like the grand finale of our algebraic performance. By dividing both sides by 2, Heidi unveiled the value of x, completing the solution. It's a perfect illustration of how mathematical operations can be strategically employed to isolate and determine the unknown. This step underscores the importance of understanding the division property of equality and its role in solving equations. It’s not just about arriving at the answer; it’s about the journey, the logical progression, and the understanding of the principles that make it all possible.

Conclusion

So, there you have it! We've dissected Heidi's solution step-by-step, justifying each move with fundamental algebraic properties. From the distributive property to the properties of equality, each step plays a crucial role in solving the equation. Remember, guys, math isn't just about finding the answer; it's about understanding the why behind it. Keep practicing, and you'll be equation-solving pros in no time! Solving equations is like following a recipe: each step is important, and when followed correctly, leads to a satisfying result. Keep practicing and soon you’ll be able to tackle even the trickiest algebraic recipes!