Solve For D In Equation: Step-by-Step Guide

by Viktoria Ivanova 44 views

Hey guys! Today, we're diving into a simple yet crucial algebraic problem: solving for a specific variable in an equation. In this case, we're tackling the equation D / (F + 2n^2) = q and our mission is to isolate D. This is a fundamental skill in mathematics and has wide applications in various fields, so let's break it down and make sure we understand each step. Trust me, once you get the hang of it, it's like riding a bike!

Understanding the Equation

Before we jump into solving, let's make sure we're all on the same page about what the equation actually means. We have D divided by the expression (F + 2n^2), and the result of this division is equal to q. Our goal is to get D all by itself on one side of the equation. To do this, we need to undo the operations that are being performed on D. In this case, D is being divided by (F + 2n^2). Remember the golden rule of algebra: whatever you do to one side of the equation, you must do to the other side to keep things balanced.

The Key Operation: Multiplication

So, how do we undo division? The inverse operation of division is multiplication! This is our key to isolating D. We need to multiply both sides of the equation by the expression (F + 2n^2). This might seem a little intimidating, but trust the process. By multiplying both sides by (F + 2n^2), we effectively cancel out the denominator on the left side of the equation, leaving us with D alone. Let's walk through the steps:

  1. Start with the original equation: D / (F + 2n^2) = q
  2. Multiply both sides by (F + 2n^2): (D / (F + 2n^2)) * (F + 2n^2) = q * (F + 2n^2)
  3. Simplify the left side: On the left side, (F + 2n^2) in the numerator and denominator cancel each other out, leaving us with just D: D = q * (F + 2n^2)

The Solution: D Isolated!

And there you have it! We've successfully isolated D. The equation now reads D = q * (F + 2n^2). This is our solution. It tells us that D is equal to the product of q and the expression (F + 2n^2). We've effectively solved for D in terms of the other variables in the equation. This is a significant achievement, guys! You've taken an equation and rearranged it to solve for a specific variable, a core skill in algebra and beyond.

Expanding the Solution (Optional)

While we've found the solution, we can take it one step further and expand the right side of the equation using the distributive property. This isn't strictly necessary, but it can sometimes be helpful depending on the context of the problem. The distributive property tells us that we can multiply q by each term inside the parentheses separately:

D = q * (F + 2n^2) D = q * F + q * (2n^2) D = qF + 2qn^2

So, we have an alternative form of the solution: D = qF + 2qn^2. Both D = q * (F + 2n^2) and D = qF + 2qn^2 are correct solutions. The choice of which form to use often depends on the specific problem or context.

Key Takeaways and Important Considerations

  • Inverse Operations: Remember that solving for a variable often involves using inverse operations. In this case, we used multiplication to undo division.
  • Balancing the Equation: The golden rule of algebra is to do the same thing to both sides of the equation. This ensures that the equation remains balanced and the solution remains valid.
  • The Distributive Property: The distributive property can be a useful tool for expanding expressions and simplifying equations.
  • Capitalization Matters: In mathematics (and in many programming languages!), capitalization matters. D is a different variable than d. Similarly, N is different from n. This is a crucial detail to keep in mind when solving equations.
  • Understanding the Goal: Always remember what you're trying to achieve. In this case, the goal was to isolate D. Keeping the goal in mind helps guide your steps.

Why This Matters: Real-World Applications

You might be wondering,