Problem 70 - Algebra A Combined Function

Question

Evaluate each expression for the given values. See Section 1.8 \(\frac{1}{2} B h ; \quad B=14\) and \(h=22\)

Step-by-Step Solution

Answer

The expression evaluates to 154 when \(B=14\) and \(h=22\).

Step 1: Substitute the Given Values

To evaluate the expression \(\frac{1}{2}Bh\), start by substituting the given values for \(B\) and \(h\). We have \(B = 14\) and \(h = 22\). This gives us: \[\frac{1}{2} \times 14 \times 22\].

Step 2: Calculate the Product

First, multiply the values of \(B\) and \(h\). That is, \(14 \times 22\). By calculating, you obtain \(308\). So, now the expression is: \[\frac{1}{2} \times 308\].

Step 3: Apply the Fraction

To apply the fraction \(\frac{1}{2}\), divide the product of \(308\) by \(2\). Thus, \(\frac{308}{2} = 154\).

Step 4: Conclusion

The value of the expression \(\frac{1}{2}Bh\) when \(B=14\) and \(h=22\) is calculated to be \(154\).

Key Concepts

Evaluating Expressions

When we talk about evaluating expressions, we are referring to the process of calculating the value of an algebraic expression by replacing its variables with specific values. This is a fundamental skill in algebra and involves simplifying the expression to a single numerical result. For instance, in the expression \( \frac{1}{2} B h \), \(B\) and \(h\) are the variables.
By assigning \(B=14\) and \(h=22\), we transform the expression into \( \frac{1}{2} \times 14 \times 22 \). This transformation allows us to proceed with the computation without the complexity of variables. Overall, evaluating expressions relies on substituting variable values and simplifying correctly through arithmetic operations.

Substitution Method

The substitution method is a key algebraic technique where you replace variables with specific numbers or expressions. It is especially useful in solving equations and evaluating expressions.
In our example, we used substitution by taking the expression \( \frac{1}{2} B h \) and replacing \(B\) and \(h\) with their given values. As a result, the expression transformed into \( \frac{1}{2} \times 14 \times 22 \).
This method helps clarify and simplify expressions, making them easier to handle. By eliminating variables through substitution, you work with simpler arithmetic expressions, paving the way for straightforward calculations.

Basic Arithmetic Operations

Basic arithmetic operations are foundational to evaluating algebraic expressions. These include addition, subtraction, multiplication, and division. In the context of our example, multiplication and division were primarily used.
First, we performed multiplication between the substituted values \(14\) and \(22\), which produced \(308\). Then, applying division—specifically using the fraction \(\frac{1}{2}\)—we divided \(308\) by \(2\), resulting in \(154\).

• Multiplication: Combines two numbers into a product, as with \(14 \times 22 = 308\).
• Division: Splits a number into equal parts, as with \(\frac{308}{2} = 154\).

Mastering these operations allows for the accurate evaluation of expressions, ensuring you can reach the correct solution efficiently.