Problem 26 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{w}{8}-4=-7$$

Step-by-Step Solution

Answer

The solution is \( w = -24 \).

Step 1: Isolate the Fraction

Start by isolating the fraction \( \frac{w}{8} \) by adding 4 to both sides of the equation. This gives us:\[ \frac{w}{8} = -7 + 4 \]Simplifying the right-hand side, we have:\[ \frac{w}{8} = -3 \]

Step 2: Solve for w

Now, to eliminate the fraction, multiply both sides of the equation by 8:\[ w = -3 \times 8 \]Simplify the right-hand side:\[ w = -24 \]

Step 3: Check the Solution

To check the solution, substitute \( w = -24 \) back into the original equation:\[ \frac{-24}{8} - 4 = -7 \]Calculate the fraction:\[ -3 - 4 = -7 \]Since both sides are equal, our solution is verified.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving equations. It involves rearranging and simplifying equations to find the unknown variable.
In our problem, we start with the equation \( \frac{w}{8} - 4 = -7 \). To find the value of \( w \), we need to isolate it on one side of the equation. We do this through a series of "undoing" actions that reverse the operations applied to \( w \).
First, we address the subtraction by adding 4 to both sides of the equation. This is a basic rule of algebra, where we perform the same operation on both sides to keep the balance of the equation intact:

• Original step: \( \frac{w}{8} - 4 = -7 \)
• Add 4 to both sides: \( \frac{w}{8} - 4 + 4 = -7 + 4 \)
• Simplified: \( \frac{w}{8} = -3 \)

This systematic approach helps us gently peel away layers of the equation until we get to the unknown variable.

Verifying Solutions

Verifying solutions is an important part of solving equations. It ensures that the value obtained is indeed the correct one and satisfies the original equation.
In the given problem, once we calculate that \( w = -24 \), we need to verify it. This is done by substituting the value back into the original equation to see if it holds true:

• Substitute \( w = -24 \) into \( \frac{w}{8} - 4 \): \( \frac{-24}{8} - 4 \)
• Simplify: \( -3 - 4 = -7 \)

If both sides are equal, as they are here, the solution is verified. This check not only confirms the correctness but also boosts confidence in the solution-solving process.
Always ensure that the manipulations and arithmetic are accurate, as a single mistake can lead to an incorrect solution.

Solving Equations with Fractions

Equations with fractions can initially seem daunting, but they can be tackled step by step with the right methods.
The equation \( \frac{w}{8} - 4 = -7 \) involves a fraction. The goal here is to eliminate the fraction to make the equation simpler. This is done by isolating the fraction term and then multiplying both sides by the denominator.
Here's how it works:

• Isolate the fraction: \( \frac{w}{8} = -3 \)
• Multiply both sides by 8 to eliminate the fraction: \( w = -3 \, \times \, 8 \)
• Resolve to get the value: \( w = -24 \)

This strategy of multiplying by the denominator is crucial because it transforms the fraction into a whole number, simplifying the equation further. Understanding this technique not only helps in solving a wide range of equations but also forms a foundational skill in algebra.