Problem 34 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$7-3-2$$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Change Subtractions to Addition of the Opposite

The expression given is \(7 - 3 - 2\). To simplify, the first step is to change each subtraction into the addition of its opposite. This gives:\[7 - 3 - 2 = 7 + (-3) + (-2)\] This means we change the minus sign in front of any number to plus and then take the negative of that number.

Step 2: Add from Left to Right

Now we must follow the order of operations to add the terms from left to right. Start by adding the first two numbers:\[7 + (-3) = 4\] This simplifies the expression to \(4 + (-2)\). Next, we add these two numbers:\[4 + (-2) = 2\] So, the original expression simplifies to 2.

Key Concepts

Order of Operations

When simplifying algebraic expressions, it's important to follow the order of operations to ensure you get the correct result. You can remember this order using the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the provided exercise, once we replace subtractions with the addition of opposites, the focus is on handling addition. Since addition and subtraction occur at the last stage of the order, we only need to ensure we perform these operations sequentially from left to right. This step-by-step processing guarantees the correct simplification of expressions like the one given. In essence, the order of operations keeps our calculations organized and accurate.

Addition of Opposites

Understanding the addition of opposites plays a crucial role in simplifying expressions involving subtraction. Instead of subtracting two numbers directly, you can turn the subtraction into an addition by considering the opposite number. For example, in the expression 7 - 3, you convert this to 7 + (-3). Here,

• The subtraction sign ( b-3 b) becomes an addition sign ( b+ b).
• The number 3 becomes its opposite, -3.

By consistently applying this technique, you can simplify each part of the expression more easily. This method harnesses the idea that subtracting a number is the same as adding its opposite. As a result, the process becomes straightforward: change subtraction into addition of the negative, then perform the operations in sequence.

Prealgebra Concepts

Prealgebra concepts are foundational ideas that prepare you for more advanced mathematical operations. They include understanding numbers, basic operations, and the properties of these operations. One essential concept in prealgebra is simplifying expressions, where we rewrite an expression in its simplest form. Simplifying expressions involves several steps:

• Recognize and translate subtraction into addition.
• Combine like terms by performing operations from left to right.
• Use arithmetic rules to ensure your work is accurate.

In the exercise, turning each subtraction into the addition of opposites taps into these concepts. It's a skill that not only helps in simplifying expressions but also deepens your understanding of negative numbers and arithmetic. Mastery of prealgebra lays the groundwork for tackling algebraic expressions efficiently, emphasizing the importance of these fundamental skills.