Problem 89 - Intermediate Algebra

Question

Simplify each numerical expression. $$ 7(6.2-7.1)-6(-1.4-2.9) $$

Step-by-Step Solution

Answer

The simplified expression is 19.5.

Step 1: Simplify Inside Parentheses

First, perform the operations inside the parentheses. For the first set of parentheses, calculate \(6.2 - 7.1\). This results in \(-0.9\). For the second set, calculate \(-1.4 - 2.9\), which results in \(-4.3\). So the expression now becomes \(7(-0.9)-6(-4.3)\).

Step 2: Multiply the Values

Next, multiply the numbers. For \(7(-0.9)\), the result is \(-6.3\). For \(6(-4.3)\), because of the negative sign in front of 6, the result is \(-6 imes -4.3 = 25.8\). So now the expression becomes \(-6.3 + 25.8\).

Step 3: Add the Results

Now add the results from Step 2: \(-6.3 + 25.8\). The result is \(19.5\).

Key Concepts

Simplification

Simplification is about making expressions easier to work with by breaking them down into simpler and more manageable pieces. In the context of numerical expressions, simplification involves performing arithmetic operations to reduce the complexity of the expression. By tackling smaller parts of the expression step by step, it becomes straightforward to reach the final simplified result. For example, in the expression given, simplifying involves completing each mathematical operation—like subtraction and multiplication—sequentially until you arrive at the simplest version of the problem.

Parentheses

Parentheses are crucial in mathematical expressions as they indicate which operations should be performed first. The rules that govern this are part of the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our exercise, dealing with the parentheses first was essential.

• First, solve the operations inside the parentheses—this prioritization prevents errors and ensures you evaluate the expression correctly.
• Calculate each set of parentheses separately before moving on to other operations. This gives us two results \(6.2 - 7.1\) yielding \(-0.9\) and \( -1.4 - 2.9\) yielding \(-4.3\) in the original expression.

Addressing the parentheses early sets a solid foundation for further steps in any mathematical problem.

Multiplication

Multiplication is the next step we tackle in simplifying numerical expressions after solving anything inside parentheses. When multiplying numbers, especially when they involve negative integers, it is important to keep in mind the rules of arithmetic with negative numbers:

• Multiplying a positive number by a negative number results in a negative product.
• Multiplying two negative numbers, on the other hand, results in a positive product.

In our example, the multiplication step involved calculating \(7 \ imes -0.9\) to get \(-6.3\), and then \(-6 \ imes -4.3\) leading to \(25.8\). This crucial step helps transition the expression from its simplified parts toward its overall reduced form.

Addition and Subtraction in Algebra

Addition and subtraction are the final steps in simplifying the expression after handling parentheses and multiplication. These operations bring together the simplified components into a single numerical value:

• When you add or subtract a series of numbers, it's often helpful to align similar terms—in this expression, we combined \(-6.3\) and \(+25.8\).
• Adding \(-6.3\) to \(25.8\) results in \(19.5\), effectively concluding the process of simplification.

Understanding the interplay between positive and negative numbers is crucial during this stage. After completing these operations, you obtain the simplest form of the expression, making it more understandable and easier to use in further calculations.