Yednia's Equation Solving Breakdown A Step-by-Step Guide
Hey guys! Today, we're diving deep into a math problem solved by our friend Yednia. Yednia tackled an equation and meticulously justified each step, which is super important in mathematics – it's not just about getting the answer, but also understanding why that answer is correct. We'll break down Yednia's process, making sure every step makes perfect sense. Let's get started!
The Equation and Yednia's Initial Setup
The equation Yednia faced was:
2 + (2x / 5) = -10
This might look a little intimidating at first, but don't worry! We'll take it one step at a time. The main goal when solving for 'x' is to isolate it – to get 'x' all by itself on one side of the equation. To do this, we need to undo all the operations that are being done to 'x', following the reverse order of operations (PEMDAS/BODMAS in reverse). Think of it like peeling back layers of an onion – we're slowly uncovering 'x'.
Step 1: Isolating the Term with 'x'
The first thing Yednia did was to get rid of the '+ 2' on the left side of the equation. Remember, we want to isolate the term with 'x', which is '(2x / 5)'. To undo adding 2, we need to subtract 2 from both sides of the equation. This is a crucial step because it maintains the balance of the equation – whatever we do to one side, we must do to the other. It's like a scale; if you take something off one side, you need to take the same amount off the other side to keep it balanced.
So, Yednia subtracted 2 from both sides:
2 + (2x / 5) - 2 = -10 - 2
This simplifies to:
(2x / 5) = -12
Why did Yednia subtract 2? The reason is the Subtraction Property of Equality. This property states that if you subtract the same value from both sides of an equation, the equation remains true. It’s a fundamental principle in algebra that allows us to manipulate equations while keeping them balanced and accurate. Yednia’s first step showcases her understanding of this property, laying the groundwork for solving the equation. By subtracting 2, she effectively isolated the term containing 'x', bringing us one step closer to finding the value of 'x'. The importance of understanding and applying these properties cannot be overstated, as they form the backbone of algebraic problem-solving.
Step 2: Eliminating the Fraction
Now, we have 2x / 5 = -12. The next step is to get rid of the fraction. We have 'x' being multiplied by 2 and then divided by 5. To undo the division by 5, we need to multiply both sides of the equation by 5. Again, we're using the principle of maintaining balance – what we do to one side, we must do to the other.
So, Yednia multiplied both sides by 5:
5 * (2x / 5) = 5 * (-12)
On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with:
2x = -60
Multiplying both sides by 5 is justified by the Multiplication Property of Equality. This property is another cornerstone of algebra, stating that multiplying both sides of an equation by the same non-zero number preserves the equality. In this context, Yednia expertly used this property to eliminate the fraction, simplifying the equation and making it easier to isolate 'x'. By multiplying by 5, she effectively undid the division, moving closer to her goal of solving for 'x'. This step not only demonstrates her procedural fluency but also her conceptual understanding of why this operation is valid and how it helps in solving the equation. Understanding these properties empowers us to manipulate equations strategically, making complex problems more manageable.
Step 3: Isolating 'x' Completely
We're almost there! We now have 2x = -60. This means 2 times 'x' equals -60. To isolate 'x', we need to undo the multiplication by 2. We do this by dividing both sides of the equation by 2.
So, Yednia divided both sides by 2:
2x / 2 = -60 / 2
This simplifies to:
x = -30
Therefore, the final answer is x = -30.
The step of dividing both sides of the equation by 2 is based on the Division Property of Equality. This property, similar to the multiplication and subtraction properties, ensures that the balance of the equation is maintained. It states that if you divide both sides of an equation by the same non-zero number, the equation remains true. Yednia’s application of this property showcases her thorough understanding of algebraic principles. By dividing both sides by 2, she successfully isolated 'x', thus revealing the solution to the equation. This final step underscores the importance of these fundamental properties in the process of solving equations, emphasizing that each operation must be justified to maintain the integrity and accuracy of the solution. In essence, Yednia’s step-by-step approach not only provides the answer but also illustrates the logical progression inherent in algebraic problem-solving.
Yednia's Justifications: The Reasons Behind the Steps
Let's recap Yednia's steps and the reasons behind them:
- Step: 2 + (2x / 5) = -10 -> (2x / 5) = -12
- Reason: Subtraction Property of Equality (Subtracting 2 from both sides)
- Step: (2x / 5) = -12 -> 2x = -60
- Reason: Multiplication Property of Equality (Multiplying both sides by 5)
- Step: 2x = -60 -> x = -30
- Reason: Division Property of Equality (Dividing both sides by 2)
These properties are the backbone of algebraic manipulation. They allow us to change the form of an equation without changing its solution. It's like rearranging the furniture in a room – you're changing the appearance, but the room itself is still the same.
Why Justification Matters in Math
You might be thinking,