Problem 35 - Prealgebra

Question

Solve each equation. Check your solution. $$2 w-4 w=-10$$

Step-by-Step Solution

Answer

The solution is \(w = 5\).

Step 1 - Simplify the Equation

Combine like terms on the left side of the equation. In this case, combine \(2w - 4w\) to get \(-2w\). The equation now becomes \(-2w = -10\).

Step 2 - Solve for w

To isolate \(w\), divide both sides of the equation by \(-2\). This results in \(w = \frac{-10}{-2}\). Simplifying this fraction gives \(w = 5\).

Step 3 - Check the Solution

Substitute \(w = 5\) back into the original equation to ensure it holds true: \(2(5) - 4(5) = 10 - 20 = -10\). Since both sides of the equation are equal, \(w = 5\) is indeed the correct solution.

Key Concepts

Solving Equations

Solving linear equations involves finding the value of the variable that makes the equation true. When you're faced with an equation, such as \(2w - 4w = -10\), the goal is to isolate \(w\) so we can know what number it represents. In step-by-step problem-solving, we simplify both sides of the equation and perform operations to isolate the variable. In this example, subtracting or combining like terms can help unravel the values that \(w\) might equal.

Combining Like Terms

Combining like terms is an essential skill when simplifying algebraic expressions. Like terms have the same variable raised to the same power.
For instance, in the problem \(2w - 4w\), both terms have \(w\) as their variable. To simplify, subtract them to get \(-2w\). By reducing equations to fewer terms, you're making it easier to see the core of the problem. Most importantly, always ensure the operation is applied properly to make the rest of the process straightforward.

Checking Solutions

Once you've found a solution, it's crucial to check if it's correct. This ensures that no mistakes were made in the earlier steps, such as combining terms or simplifying the equation wrongly.
In our example, we suspected \(w = 5\) to be the answer. Substituting it back into the original equation \(2(5) - 4(5)\) gives \(10 - 20 = -10\), which equals the right side of the original equation.

• If both sides of the equation match, the solution is verified.
• If not, review your work for potential errors.

Checking prevents errors from being carried forward into future calculations.

Prealgebra

Prealgebra lays the groundwork for understanding basic algebraic principles such as variable manipulation and equation solving.
The goal here is to get comfortable with balancing equations, understanding variables, and knowing how to perform arithmetic operations across equations.

• Balancing is crucial: whatever operation is done on one side must be mirrored on the other.
• Knowing when and how to combine terms or divide to isolate a variable is a fundamental skill.

This foundational knowledge prepares students for more complex algebraic operations they will encounter in further mathematics courses or everyday problem-solving scenarios.