Problem 30 - Prealgebra

Question

Solve each equation. Check your solution. $$-116=-4 w$$

Step-by-Step Solution

Answer

The solution to the equation is \(w = 29\).

Step 1: Isolate the Variable

The equation given is \(-116 = -4w\). To solve for \(w\), we need to isolate \(w\) by getting rid of the coefficient \(-4\). Since \(-4\) is multiplied by \(w\), we can do the opposite operation, which is division, to both sides to isolate \(w\). Divide both sides of the equation by \(-4\) to obtain:\[-116 \div -4 = w\].

Step 2: Perform the Division

Proceed with the division from Step 1: \[-116 \div -4 = 29\]. Thus, we find that \(w = 29\).

Step 3: Verify the Solution

In order to ensure that the solution is correct, substitute \(w = 29\) back into the original equation. The original equation is \(-116 = -4w\). Substitute \(w = 29\) into the equation: \[-116 = -4(29)\]. Now calculate the right-hand side: \(-4 \times 29 = -116\). Since both sides are equal, the solution is verified.

Key Concepts

Checking Solutions

Once you've found a potential solution to an equation, the next step is verification. Verification means plugging the solution back into the original equation to ensure that it satisfies the equation. This step ensures that you've correctly solved for the variable and haven't made an error in calculation.

Here's how you can verify:

• Take the solution you found, in this case, \(w = 29\).
• Replace the variable \(w\) in the original equation with your solution.
• Re-evaluate the equation to check if both sides are equal.

In the example with the equation \(-116 = -4w\), substituting \(w = 29\) gives us \(-116 = -4 \times 29\). Calculating the right side gives \(-116\), confirming the solution. Verification reassures correctness and builds confidence in your math skills.

Isolating Variables

Isolating the variable in an equation is key to solving it. This means getting the variable by itself on one side of the equation. Think of it as "unwrapping" the variable from any operations or numbers that are attached to it.

For example:

• Identify the operation affecting the variable. In the equation \(-116 = -4w\), the variable \(w\) is multiplied by \(-4\).
• To isolate \(w\), perform the inverse operation. Here, that means dividing by \(-4\).
• Apply this operation to both sides of the equation to maintain the balance. So you divide \(-116\) by \(-4\), resulting in \(w = 29\).

The goal is to have the variable alone on one side, making it easy to see the value of the variable.

Division in Equations

When solving equations, division often comes into play, especially when dealing with coefficients. Division helps in isolating the variable when it has been multiplied by a number. To keep equations balanced, any operation performed on one side must be performed on the other as well.

In the given equation, you need to:

• Recognize that \(w\) is multiplied by \(-4\).
• Use division as the inverse operation to multiplication. Divide both sides by \(-4\) to isolate \(w\).
• Calculate the division: \(-116 \div -4\). This gives \(w = 29\).

This process of dividing allows us to solve equations efficiently and correctly. Remember, division is a powerful tool in making complex equations simple and manageable.