Question

Evaluate the expression. $$12-9+7$$

Short answer

The evaluation of the expression \(12 - 9 + 7\) is 10.

Step-by-step solution

Perform the subtraction

Subtract 9 from 12 following the rule of 'Addition and Subtraction' from left to right. This implies the operation \(12 - 9\), which equals 3.

Perform the addition

Add 7 to the result of Step 1. The operation now is \(3 + 7\), which equals 10.

Key concepts

Order of Operations

Understanding the order of operations is crucial when it comes to evaluating mathematical expressions accurately. It refers to the specific sequence in which operations should be performed within a math problem. The general rule is expressed by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). We start by solving expressions inside the parentheses, then move onto exponents. Next comes multiplication and division, which are treated with equal importance and are done from left to right. Finally, we tackle additions and subtractions, again moving from left to right.

It's vital for students to remember this sequence because neglecting the proper order can lead to incorrect results. For instance, in the given problem \(12-9+7\), there are no parentheses, exponents, or multiplication and division operations. Therefore, the focus is solely on the last part of PEMDAS: addition and subtraction. Since both operations are of equal precedence, we simply work from left to right as shown in the solution steps.

Arithmetic Operations

The foundation of evaluating expressions lies in the four basic arithmetic operations which are: addition, subtraction, multiplication, and division. These operations are the building blocks for more complex mathematical concepts. When we add numbers, we are combining two or more quantities to find their total. Subtraction is the process of finding the difference between numbers, essentially determining how much one quantity is less than another.

Multiplication involves combining equal groups to find out how many items there are in total, while division is the process of splitting a quantity into equal parts. In the provided example, we use subtraction (\(12 - 9\)) to find the difference between 12 and 9, then addition (\(3 + 7\)) to combine the result with 7, following these basic operations orderly.

Basic Algebra

Moving beyond arithmetic, we enter the realm of basic algebra, a branch of mathematics that utilizes letters and symbols to represent numbers and quantities in equations and expressions. Algebra enables us to generalize arithmetic by using variable placeholders that can stand for unknown values. This paves the way to solving problems that are more abstract than simple number manipulation.

When dealing with expressions as in our original exercise, 'evaluate the expression', it is a matter of applying arithmetic operations to find a numerical result. The provided example might not explicitly showcase traditional algebra with variables, but it does set the stage for the logical reasoning and manipulation of numbers that are fundamental in algebra. As students progress, they'll encounter expressions that involve variables and more complex operations, but the groundwork of understanding how to evaluate expressions using the proper rules and procedures remains the same.