Problem 36 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ \frac{a}{8}-2 a+1, \text { if } a=24 $$

Step-by-Step Solution

Answer

The expression evaluates to -44.

Step 1: Substitute the Given Value

The first step is to substitute the given value of \( a \) into the expression. The expression is \( \frac{a}{8} - 2a + 1 \) and we are told that \( a = 24 \). So substitute 24 in place of \( a \): \( \frac{24}{8} - 2(24) + 1 \).

Step 2: Simplify the Fraction

Now, simplify the fraction \( \frac{24}{8} \). Dividing 24 by 8 gives: \( \frac{24}{8} = 3 \).

Step 3: Perform the Multiplication

Next, perform the multiplication: \(-2 \times 24\). Multiply \(-2\) by 24, which gives: \(-2 \times 24 = -48 \).

Step 4: Combine All Terms

Combine the results from the previous steps to form a single expression. Add \(3\), subtract \(48\), and add \(1\): \(3 - 48 + 1\).

Step 5: Simplify the Expression

Finally, simplify the expression by combining the numbers. Start by calculating \(3 - 48\), which gives: \(3 - 48 = -45\). Then add \(1\) to \(-45\): \(-45 + 1 = -44\).

Step 6: Conclusion

The simplified answer for the expression \( \frac{a}{8} - 2a + 1 \) when \( a = 24 \) is \(-44\).

Key Concepts

Substitution in Expressions

Substitution means replacing a variable with a given number. This is one of the fundamental steps in evaluating algebraic expressions, particularly when a specific value for the variable is provided. In our example, the expression is \( \frac{a}{8} - 2a + 1 \), and we need to find its value when \( a = 24 \).

When substituting, simply plug the given number in place of the variable. So, replace \( a \) with 24:

• Replace \( a \) with 24, resulting in \( \frac{24}{8} - 2(24) + 1 \).

This gives us numbers to work with, making it easier to simplify the expression afterwards.

Simplifying Expressions

Simplifying algebraic expressions is about making them as straightforward as possible. After substitution, the expression \( \frac{24}{8} - 2(24) + 1 \) is what we need to simplify.

First, take care of operations one step at a time, following the order of operations:

• Start with division: \( \frac{24}{8} = 3 \).
• Next, handle multiplication: \(-2 \times 24 = -48 \).
• Finally, combine all these numbers together: \(3 - 48 + 1 \).

This systematic approach allows us to break down what initially looks complex into simpler parts.

Fractions in Algebra

Fractions are often encountered in algebra, and knowing how to simplify them is crucial. In our example, the fraction \( \frac{24}{8} \) appears right after substituting values.

Simplifying fractions involves dividing the numerator by the denominator:

• Divide 24 by 8 to get \( 3 \).

This reduces any fractional component to a whole number, making the rest of the expression easier to tackle.

Fractions will often appear in various algebra problems, but the process of simplifying them remains the same.

Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division are essential in solving algebraic expressions. Once we substitute and simplify a fraction, we're left with basic arithmetic.

Here's how it works with our example:

• Perform multiplication: \(-2 \times 24 = -48 \).
• Then add and subtract in sequence: First \(3 - 48 = -45 \), and then add \(1\) to get \(-44 \).

This involves following the order of operations, often remembered with PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Mastering these operations is key as they form the building blocks of more complex algebraic expressions.