Problem 4 - Intermediate Algebra

Question

Find the value of each algebraic expression at the given replacement values. See Examples 1 and 2 \(7.1 a\) when \(a=1.5\)

Step-by-Step Solution

Answer

The value of the expression is 10.65.

Step 1: Identify the Algebraic Expression and Given Value

The algebraic expression given is \(7.1a\). We need to find its value when \(a = 1.5\).

Step 2: Substitute the Given Value into the Expression

Replace \(a\) with 1.5 in the expression \(7.1a\). This gives us: \(7.1 \times 1.5\).

Step 3: Calculate the Product

Multiply \(7.1\) by \(1.5\). This calculation can be solved as follows: \(7.1 \times 1.5 = 10.65\).

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can consist of numbers, variables (like \(a\), \(b\), or \(x\)), and operation symbols such as addition, subtraction, multiplication, and division. For instance, in the expression \(7.1a\), "7.1" is known as a coefficient, and "a" is the variable. Coefficients are the numerical part of the terms and tell us how much of the variable we have.

It is essential to be comfortable identifying these components because they form the foundation for manipulating expressions and equations. In algebra, variables represent unknown or changeable numbers, and you can substitute different values to evaluate the expressions. Understanding how to use algebraic expressions is a core skill in mathematics that allows more complex problem-solving and enables us to generalize mathematical relationships.

Substitution in Algebra

Substitution is the process of replacing a variable in an algebraic expression with a given value. This helps in simplifying expressions and finding numerical solutions. Take the expression \(7.1a\). If we know that \(a = 1.5\), substitution means replacing "a" with "1.5".

What you do is take every instance of the variable "a" and replace it with the number 1.5. So, the expression changes from \(7.1a\) to \(7.1 \times 1.5\). This is a crucial step in algebra because it allows us to transform abstract expressions into concrete values that can be calculated.

• Identify the variable and its value
• Replace the variable with the given number

This approach is fundamental in algebraic problem-solving and helps you evaluate expressions accurately with any variable and its value.

Multiplication of Decimals

Multiplying decimals involves the same basic steps as multiplying whole numbers, with a slight addition when placing the decimal in the product. Let's consider multiplying \(7.1\) by \(1.5\).

First, ignore the decimals and multiply \(71\) by \(15\) as if they were whole numbers. The result would be \(1065\). Next, count the total number of decimal places in the factors. Here, there is one decimal place in each factor, which sums to two decimal places.

• Multiply as if there are no decimals
• Count the decimal places in both numbers
• Place the decimal in the result: move two places from the right

Thus, you place the decimal in \(1065\) to get \(10.65\). This approach ensures precision when dealing with decimals in algebra, helping to arrive at the correct solution. Multiplying decimals accurately is crucial, particularly in scientific and financial calculations.