Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the variable. \(\frac{1}{4}\left(8 w-4^{2}\right)\) for \(w=1\)

Step-by-Step Solution

Answer

The solution is -2.

Step 1: Substitute the given value

Substitute the given value of the variable into the expression. For \( w = 1 \), replace \( w \) in the expression \( \frac{1}{4}(8w - 4^2) \) with 1. The expression becomes \( \frac{1}{4}(8 \times 1 - 4^2) \).

Step 2: Calculate the power

Calculate \( 4^2 \). Since \( 4^2 = 16 \), substitute this value back into the expression. The expression is now \( \frac{1}{4}(8 \times 1 - 16) \).

Step 3: Simplify inside the parentheses

Perform the multiplication \( 8 \times 1 \) which equals 8. Then, subtract 16 from this result: \( 8 - 16 = -8 \). The expression simplifies to \( \frac{1}{4}(-8) \).

Step 4: Perform the division

Divide \(-8\) by 4. The result of this division is \(-2\).

Key Concepts

Substitution Method

The substitution method is a critical technique in algebra where you replace a variable with a given value. This is often used to solve equations and evaluate expressions. In our example, we have an expression with a variable: \[ \frac{1}{4}(8w - 4^2) \] Given that \( w = 1 \), we substitute the value of \( w \) into the expression. This changes the variable expression to a numerical one. It looks like this: \[ \frac{1}{4}(8 \times 1 - 4^2) \] Substitution is very straightforward and allows you to transform the expression into something you can calculate. Always ensure the substitution is done correctly, as any mistake here can lead to errors in your final answer.

Evaluating Expressions

Evaluating an expression means performing all the necessary operations to arrive at a final numerical result. After substitution, the expression \( \frac{1}{4}(8 \times 1 - 4^2) \) needs to be simplified step by step.

• Calculate the exponent: The first step involves finding the value of the power, \( 4^2 \). This equals \( 16 \).
• Simplify inside the parentheses: Replace \( 4^2 \) with \( 16 \), then perform the multiplication \( 8 \times 1 \), which is \( 8 \). Finally, subtract \( 16 \) from \( 8 \) to get \(-8\).

At this point, the expression is simplified to: \[ \frac{1}{4}(-8) \] Completing these steps accurately ensures that the expression is correct and ready for the final calculation.

Multiplying and Dividing Fractions

Dealing with fractions involves understanding both multiplication and division. In this context, you multiply \(-8\) by \( \frac{1}{4} \). Here's how:

• Multiplying a fraction by a number: To multiply a fraction by an integer, take the number and multiply it with the numerator (the top part of the fraction). In \( \frac{1}{4}(-8) \), multiply \(-8\) by \(1\), which gives \(-8\).
• Performing the division: Now divide \(-8\) by the denominator, \(4\). Doing this, you get \(-2\).

Understanding how to multiply and divide with fractions is vital, especially when dealing with algebraic expressions involving fractions. Practice consistently to master these core steps.