Problem 13 - Elementary Algebra

Question

Find the sums. 35+(-78)

Step-by-Step Solution

Answer

Answer: -43

Step 1: Identify the signs of the numbers

We have two numbers: 35 (a positive number) and -78 (a negative number). When adding numbers with different signs, we need to subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

Step 2: Find the absolute values of the numbers

The absolute value of a number is its distance from zero, without considering its sign. In this case, the absolute values are as follows: Absolute value of 35: |35| = 35 Absolute value of -78: |-78| = 78

Step 3: Determine which number has the larger absolute value

Comparing the absolute values, we can see that |-78| = 78 is larger than |35| = 35. So, the larger absolute value is 78, and the sign of our final answer will be negative, since -78 has a negative sign.

Step 4: Subtract the smaller absolute value from the larger absolute value

Now, we will subtract the smaller absolute value (35) from the larger absolute value (78): 78 - 35 = 43

Step 5: Assign the sign to the result

Since the number with the larger absolute value (-78) has a negative sign, our final result will also have a negative sign. Therefore, the sum of 35 and -78 is: 35 + (-78) = -43

Key Concepts

Absolute Value

Understanding absolute value is crucial in mathematics, especially when dealing with integers. The absolute value of a number is the distance between that number and zero on the number line, without considering the direction. In simpler terms, it's the number's 'numerical' value irrespective of its sign.

For example, both -5 and 5 are five units away from zero. Thus, their absolute value is the same: |(-5)| = |5| = 5. Knowing this concept helps us to solve various algebraic problems, particularly when we are required to compare sizes or work with negative numbers.

Negative Numbers

Negative numbers are integers that are less than zero. They are represented with a minus sign (-) in front. In the context of everyday life, negative numbers denote a lack of something, such as a debt or below-zero temperatures.

In mathematical operations, negative numbers affect the outcome depending on the operation being performed. For instance, when adding a negative number to a positive one, you essentially subtract its absolute value from the positive number, as shown in the solved exercise: 35 + (-78) results in a subtraction: 35 - 78.

Integer Addition

Addition of integers involves combining positive and negative whole numbers. The rules are straightforward: If the signs are the same, you simply add the absolute values and keep the sign. However, with different signs, you find the difference between the absolute values and keep the sign of the number with the larger absolute value.

In the exercise given, 35 + (-78), we add a positive number to a negative number, which calls for subtracting the smaller absolute value from the larger one and preserving the sign of the number with the larger absolute value. Understanding integer addition helps students navigate through more complex algebraic expressions with confidence.

Algebraic Operations

Algebraic operations are the foundation of algebra and involve operations like addition, subtraction, multiplication, and division applied to algebraic expressions and equations. Addition, as covered in the exercise, is one of the most basic operations. It allows us to combine quantities and understand how they relate to each other in an expression.

Performing algebraic operations with integers, particularly, requires an understanding of the properties of numbers and their signs to correctly approach problems. In the given example, combining integer addition with knowledge of absolute values and negative numbers enables us to find the sum of a positive and a negative integer. Mastery of these fundamental operations is essential for advancing in mathematics and problem-solving. Remember that a thorough concept grasp and practice are key to becoming proficient in algebraic operations.