Problem 70 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$-\frac{x}{2}=3$$

Step-by-Step Solution

Answer

The solution to the equation \(-\frac{x}{2} = 3\) is \(x = -6\).

Step 1: Isolate x by using inverse operations

Currently, \(x\) is being divided by 2. To isolate \(x\), you need to do the opposite operation, which is multiplication. Multiply both sides of the equation by -2 to cancel out dividing by -2 on the left-hand side of the equation. This results in \(x = -2 * 3\).

Step 2: Solve for x

Multiplying -2 and 3 gives -6, so the solution to the equation is \(x = -6\).

Step 3: Check the solution

Substitute \(x = -6\) back into the original equation \(-\frac{x}{2} = 3\). If the left-hand side equals the right-hand side, then the solution is correct. Substituting gives \(-\frac{-6}{2}\). This simplifies to 3, which is indeed the right-hand side of the original equation, so the solution is correct.

Key Concepts

Inverse Operations

In algebra, mastering inverse operations is essential for solving equations. What does this mean? Simply put, inverse operations are pairs of mathematical operations that undo each other. Some common examples include addition and subtraction, and multiplication and division.

Let's look at the example equation given: \[ -\frac{x}{2} = 3 \.\] To solve for \(x\), you need to 'undo' the division by 2. How? By using its inverse operation: multiplication. It's like a mathematical magic trick—applying multiplication here will help the division 'disappear'. So, when you multiply both sides by -2, the equation balances out, and you're a step closer to finding \(x\).

This step isn't just about calculations; it's about understanding the relationship between operations. By recognizing which operations are inverses of each other, you can strategically apply them to clear the path towards the solution.

Isolate Variables

The goal of isolating the variable in an algebraic equation is much like finding a hidden treasure by following a map. The 'X' marks the spot, and in algebra, getting \(x\) alone on one side of the equals sign leads you to the treasure—your solution.

In our example problem, \(x\) starts out trapped: \[ -\frac{x}{2} = 3 \.\] To free \(x\) and isolate it, we must remove all other numbers and operations from its side of the equation by using inverse operations—like reversing the steps of a dance until you're back where you started.

The process is a delicate balance, though. Whatever we do to one side, we must do to the other to maintain equality. It's like track scales—the weights must be even on both sides. By multiplying both sides of the equation by -2, we're able to isolate \(x\) effectively, showing the path to \(x = -6\).

Why Isolate?

Isolating the variable grants clarity. You transform a complex statement into a simple 'This equals that'—the essence of an algebraic solution.

Equation Solutions

Discovering the solutions to equations is like cracking a code. You've been given a complex series of symbols, and your task is to simplify them down to an understandable message—in this case, the value of the unknown variable, usually represented by \(x\).

In our textbook example, the solution received through inverse operations and variable isolation is \(x = -6\). But our work isn't done with just finding this number. True understanding—and assurance that we haven't been led astray—comes from verifying our answer.

This verification step is akin to checking your work in a puzzle. We take our solution, \(-6\), and place it back into the original equation to see if it fits perfectly. By substituting \(x\) with \(-6\), we're testing the equation's integrity. If it holds true—if the left side equals the right as it does here with 3 equals 3—then we can declare: 'Eureka! The solution is correct.'

The Satisfaction of a Correct Solution

Nothing beats the satisfaction of not only finding but also confirming that your solution to an equation is correct. It's the final piece of the puzzle being snugly fitted into place, completing the beautiful picture of your algebraic endeavor.