Problem 67 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. Evaluate \(A=\frac{1}{2} h(a+b)\) for \(a=10, b=16,\) and \(h=7\)

Step-by-Step Solution

Answer

So, substituting \(a=10, b=16,\) and \(h=7\) into \(A=\frac{1}{2} h(a+b)\), we get \(A= 91\).

Step 1: Substituting the values

Plug the values of \(a=10, b=16,\) and \(h=7\) into the formula \(A=\frac{1}{2} h(a+b)\), we get \(A=\frac{1}{2}(7)(10+16)\).

Step 2: Simplifying inside the parentheses

Solving for \(10+16\) inside the parentheses first, as per operational order, we get \(A=\frac{1}{2}(7)(26)\).

Step 3: Further Simplification

Now we perform the multiplication. So, \(A=\frac{1}{2}(182)\).

Step 4: Final Calculation

Finally we perform the remaining multiplication: \(A=\frac{1}{2}\times 182 = 91\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique used to evaluate algebraic expressions. It involves replacing variables with their corresponding numerical values. For example, if you have an expression such as \( A = \frac{1}{2} h(a+b) \) , and you are given the values of the variables a, b, and h, your first step is to substitute those given values into the expression.

In the exercise provided, we are given that \( a=10 \text{, } b=16 \text{, and } h=7 \) . The process of evaluating \( A \) starts by plugging these numbers into the original formula: \begin{align*} A &= \frac{1}{2}\times7\times(10+16)\text{.}\text{This step transforms the algebraic expression} & \text{into an arithmetic problem, making it possible}\text{to calculate the actual value of} A.\end{align*}
Accurate substitution is critical for obtaining the correct answer and requires careful attention to ensure that each variable is replaced with the correct value. This process helps in simplifying complex problems and sets the stage for further computations.

Operational Order in Algebra

Algebraic expressions often involve multiple operations such as addition, subtraction, multiplication, and division. The operational order, commonly known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right), dictates the sequence in which these operations should be performed. Adherence to this order ensures that the expression is simplified correctly.

In the example given, after substituting the values, you are faced with the expression \( \frac{1}{2}(7)(10+16) \) . Before multiplying by \frac{1}{2} and 7, you must first address the operation within the parentheses (10+16), as parentheses have the highest priority in the order of operations. This leads to a new, simpler expression: \( A = \frac{1}{2}(7)(26) \) .

Only after simplifying the expression within the parentheses should you move on to multiplication and division, strictly following the predetermined operational order to avoid any errors in your final calculation.

Simplifying Expressions

Simplifying algebraic expressions is the process of reducing them to their simplest form. It can involve combining like terms, using algebraic properties such as distributive, associative, and commutative properties, and following the operational order as discussed earlier.

In our example, \( A = \frac{1}{2}(7)(26) \) is to be simplified by first performing the multiplication within the parentheses. Multiplying 7 by 26 yields 182, resulting in \( A = \frac{1}{2}(182) \) . The final step is to multiply 182 by \frac{1}{2}, which simplifies to 91. Hence, the expression \( A = 91 \) is the simplest form of the original algebraic expression.

Simplifying an expression makes it more understandable and easier to work with, especially when dealing with complex equations or larger numbers. By breaking down the process into clear, manageable steps, we ensure that each part of the expression is addressed methodically, leading to an accurate and simplified result.