Problem 61 - Elementary Algebra

Question

Find the sums for the the following problems. \(-373+(-14)\)

Step-by-Step Solution

Answer

Answer: The sum of -373 and -14 is -387.

Step 1: Identify the given numbers

We are given the numbers \(-373\) and \(-14\).

Step 2: Find the sum of the numbers

To find the sum, we simply add the two numbers together: \((-373) + (-14) = -373 - 14\)

Step 3: Subtract the numbers

Now we have \(-373 - 14\). Since we are subtracting a negative number, this means we move further along the negative axis. Therefore, we can add their absolute values together: \(373 + 14 = 387\)

Step 4: Write the final answer

Since we found that their sum is equal to 387, and as we mentioned earlier, the result should be negative, the final answer is: \(-373 + (-14) = \boxed{-387}\).

Key Concepts

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the properties and manipulation of numbers. It plays a critical role in preparing students for advanced mathematics and understanding complex number relationships.

When it comes to adding negative numbers, such as (-373) + (-14), it's essential to visualize the number line. In algebra, negative numbers are represented to the left of zero on this line. So, when we add two negative numbers, we're essentially moving further left, indicating a larger negative value. In this example, we combine the magnitudes of both numbers and then apply the negative sign to represent their position on the number line.

Negative Number Operations

Working with negative numbers is a fundamental skill in arithmetic, which can sometimes be counterintuitive. For instance, when we talk about adding a negative number to another negative number, it's helpful to think about combining losses or debts, where the total loss or debt increases.

Using our textbook problem as an example, adding (-373) and (-14) together is akin to having one debt of \(373 and another of \)14, resulting in a total debt of $387, which in number terms is expressed as -387. It highlights the key concept that adding negative numbers results in a 'more negative' sum.

Arithmetic with Integers

Arithmetic with integers involves the basic operations of addition, subtraction, multiplication, and division. But, it gets slightly trickier when dealing with negative integers.

Remember, the rules change when we include negatives: adding a negative number is the same as subtracting its positive counterpart. This rule simplified step 3 of our problem, turning (-373) + (-14) into the subtraction of positive numbers 373 + 14. Once computed, the negative sign is appended to reflect the correct value of -387, indicating our result lies to the left of zero on the number line.

Absolute Value

The absolute value of a number is its distance from zero on the number line, disregarding its direction (positive or negative). For any number 'x', its absolute value is denoted as |x|.

When we added -373 + (-14) in our exercise, we actually added their absolute values: |373| + |14|, which equals 387. The absolute value is always non-negative. This concept helps us separate the magnitude of a number from its sign, allowing for simpler calculations before applying the correct sign to our result. Thus, the proper notation for our exercise's final answer would be |-387|, confirming that our sum has a magnitude of 387 units away from zero.