Problem 52 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ 2+7-10+2 $$

Step-by-Step Solution

Answer

Answer: 1

Step 1: Add 2 and 7

To begin, we will add 2 and 7, which results in 9: $$ (2+7) - 10 + 2 = 9 - 10 + 2 $$

Step 2: Subtract 10 from 9

Next, we will subtract 10 from 9, which results in -1: $$ 9 - (10) + 2 = -1 + 2 $$

Step 3: Add 2 to -1

Finally, we will add 2 to -1, which results in 1: $$ (-1) + 2 = 1 $$ So, the value of the expression is 1.

Key Concepts

Adding and Subtracting Integers

Understanding how to work with integers—which are positive and negative whole numbers—is essential in basic algebra. When adding integers, if the signs are the same, we add the absolute values and keep the sign. For example, adding two negative integers like \( -3 + (-2) \) gives us \( -5 \). Conversely, when the signs are different, as in \( -3 + 2 \) or \( 3 + (-2) \), we subtract the smaller absolute value from the larger one and keep the sign of the larger absolute value, resulting in \( -1 \) and \( 1 \) respectively.

Subtracting integers, on the other hand, can be thought of as adding the opposite. The operation \( a - b \) is the same as \( a + (-b) \). So, if we have to solve \( 7 - 10 \), we would change it to \( 7 + (-10) \) and proceed with adding integers of different signs. In our original exercise, the process involved both adding and subtracting integers to arrive at the final answer.

Order of Operations

The order of operations is a fundamental concept that dictates the sequence in which multiple operations should be performed to correctly evaluate an expression. The conventional order, often remembered by the acronym PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

This helps avoid confusion and ensures consistency in solving mathematical problems. For instance, if an expression contains both addition and subtraction, like \(2 + 7 - 10 + 2\), we work from left to right, dealing with addition and subtraction as they appear. It's crucial that operations within parentheses are completed first, but since there are none in our example, we proceed directly with addition and subtraction in order.

Evaluating Expressions

Evaluating expressions involves finding the numerical value of an algebraic expression by substituting the values of variables and performing the operations in the correct order. The skill of evaluating expressions is key to understanding and solving algebraic equations and simplifying expressions.

When dealing with simple numerical expressions, like in our exercise \(2 + 7 - 10 + 2\), we apply the order of operations even if only basic operations are involved. No parentheses, exponents, or multiplication and division means that addition and subtraction are performed as they appear from left to right. Each step simplifies the expression further until we arrive at the final numeric answer, which, as demonstrated in the solution steps, is 1.