Problem 57 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{3}{8} w=5$$

Step-by-Step Solution

Answer

The solution is \( w = \frac{40}{3} \).

Step 1: Isolate the variable

To solve for the variable \( w \), we need to isolate it on one side of the equation. Currently, the equation is \( \frac{3}{8}w = 5 \). We can isolate \( w \) by multiplying both sides of the equation by the reciprocal of \( \frac{3}{8} \), which is \( \frac{8}{3} \).

Step 2: Perform the multiplication

Multiply both sides of the equation by \( \frac{8}{3} \):\[ \frac{8}{3} \times \frac{3}{8}w = 5 \times \frac{8}{3} \]On the left side, the fractions \( \frac{8}{3} \) and \( \frac{3}{8} \) cancel each other out, leaving us with \( w \). On the right side:\[ w = \frac{40}{3} \]

Step 3: Check the solution

Substitute \( w = \frac{40}{3} \) back into the original equation to ensure it satisfies it. The original equation is \( \frac{3}{8} w = 5 \).Substitute and simplify:\[ \frac{3}{8} \times \frac{40}{3} = 5 \]Simplify the left side:\[ \frac{3 \times 40}{8 \times 3} = 5 \]\[ \frac{40}{8} = 5 \]\[ 5 = 5 \]Since both sides of the equation are equal, \( w = \frac{40}{3} \) is indeed the correct solution.

Key Concepts

Isolating the Variable

In solving equations, the first step often involves isolating the variable. This means getting the variable by itself on one side of the equation. For the equation \( \frac{3}{8} w = 5 \), the goal is to "free" \( w \) from the fraction. A variable can be isolated by performing operations that "undo" what's currently being done to the variable. Here, \( w \) is multiplied by \( \frac{3}{8} \).
To isolate \( w \), multiply both sides of the equation by the reciprocal of \( \frac{3}{8} \), which is \( \frac{8}{3} \).

This operation effectively cancels out \( \frac{3}{8} \) thanks to the reciprocal property, because \( \frac{8}{3} \times \frac{3}{8} = 1 \). When you perform this operation, \( w \) stands alone:

• On the left: \( \frac{8}{3} \times \frac{3}{8} w \rightarrow w \)
• On the right: \( 5 \times \frac{8}{3} = \frac{40}{3} \)

This lead us to the solution \( w = \frac{40}{3} \). Isolating the variable is crucial, as it is the primary technique for solving linear equations.

Checking Solutions

Checking your solutions is a vital step when solving equations. It ensures that the solution you obtained is correct. After you believe you've solved the equation, substitute your solution back into the original equation to see if it holds true. For our solution, we found that \( w = \frac{40}{3} \).
To check:

• Replace \( w \) in the original equation \( \frac{3}{8} w = 5 \) with \( \frac{40}{3} \).
• The equation becomes: \( \frac{3}{8} \times \frac{40}{3} \).
• Calculate: \( \frac{3 \times 40}{8 \times 3} \rightarrow \frac{120}{24} = 5 \).

Since both sides of the equation are equal when using \( w = \frac{40}{3} \), you can confidently conclude that your solution is correct. This step helps avoid any potential mistakes and confirms that your solution satisfies the original equation.

Fractions in Equations

Fractions in equations can sometimes make the solving process appear complicated, but understanding how to handle them simplifies the work. In our exercise, we encountered the fraction \( \frac{3}{8} \) in the equation \( \frac{3}{8} w = 5 \).
Key strategies to work with fractions in equations:

• Reciprocal Multiplication: To eliminate the fraction, use the concept of reciprocals. Multiply both sides by the reciprocal of the fraction to clear it. The reciprocal is essentially the fraction "flipped"—if you have \( \frac{3}{8} \), its reciprocal is \( \frac{8}{3} \).
• Balancing Equations: Whatever you do to one side of the equation, do to the other. This maintains equality.
• Simplification: Once fractions are cleared, simplify the remaining expression to get to the solution.

Understanding how to manipulate fractions within equations is key to not feeling overwhelmed by their presence. Mastering these strategies will help you tackle equations with fractions more confidently.