Problem 14 - Elementary Algebra

Question

Add and subtract. $$ 2-7+(-9) $$

Step-by-Step Solution

Answer

The result is -14.

Step 1: Understand the Problem

The expression we need to evaluate is \( 2 - 7 + (-9) \). This involves adding and subtracting integers.

Step 2: Perform the First Subtraction

First, subtract 7 from 2: \( 2 - 7 = -5 \). The result is a negative number because we're subtracting a larger number from a smaller one.

Step 3: Add the Negative Number

Now, add \(-9\) to the result from Step 2: \( -5 + (-9) \). When adding a negative number, it's equivalent to subtraction.

Step 4: Calculate the Final Result

Add the two negative numbers: \(-5 + (-9) = -5 - 9 = -14\). We combine them to get the final answer.

Key Concepts

Adding Integers

Adding integers is a fundamental concept in mathematics that you use every day without realizing it. To add integers, you need to keep track of their signs—positive or negative. Here’s how it works step-by-step:

• When adding two positive integers, simply add their absolute values. For example, to calculate \( 3 + 5 \), you add them together to get 8.
• When adding two negative integers, add their absolute values, but keep the negative sign in the result. For instance, \((-2) + (-4)\) becomes \(-(2 + 4) = -6\).
• If one integer is positive and the other is negative, find the difference between their absolute values and keep the sign of the larger absolute value. For example, \(5 + (-3)\) means we subtract 3 from 5 to get 2, which remains positive.

Keeping these simple rules in mind can help you handle any integer addition, whether they are positive, negative, or a mix of both. Remember, practice is key!

Subtracting Integers

Subtracting integers might seem tricky at first, but it becomes intuitive once you grasp the concept of adding the opposite. Subtraction can often be turned into addition, simplifying the process.

• To subtract a positive integer, add its opposite (negative). For example, \(7 - 3\) is the same as \(7 + (-3)\). This results in 4, because we subtract 3 from 7.
• When subtracting a negative integer, you actually add the absolute value. So, \(5 - (-2)\) becomes \(5 + 2\), giving you 7.
• If you are working with two negative integers, like \(-5 - (-3)\), transform it to \(-5 + 3\), which simplifies to \(-2\) by adding the absolute value of 3 to -5.

This approach to subtraction is extremely helpful in simplifying complex problems and preventing common errors, especially when dealing with multiple integers in one expression.

Negative Numbers

Negative numbers are an important tool in mathematics, representing values below zero. Understanding how they behave in arithmetic operations is crucial.

• Negative numbers are key in representing debts, temperatures below freezing, or directions below a baseline.
• When adding negative numbers, think of it as moving left on a number line. Thus, \(-3 + (-5)\) means you move 5 steps left from -3, ending at -8.
• Subtracting a negative number is like adding a positive one, as reversing a debt is akin to gaining money. For instance, \(-7 - (-2)\) becomes \(-7 + 2\), resulting in -5.

Understanding the properties and operations of negative numbers can help you solve both simple and complex problems while avoiding mistakes. Practice handling negative numbers, and they will become second nature.