Problem 63 - Elementary Algebra

Question

Find the sums for the the following problems. \(-47.03+(-22.71)\)

Step-by-Step Solution

Answer

Answer: The sum of -47.03 and -22.71 is -69.74.

Step 1: Add the absolute values.

First, ignore the negative signs and find the sum of the absolute values of the two numbers. In this case, add 47.03 and 22.71 together. \(47.03 + 22.71 = 69.74\)

Step 2: Assign the negative sign to the result.

Since both original numbers were negative, the sum will also be negative. Therefore, assign a negative sign to the result we found in step 1. \(-69.74\)

Step 3: Write the final answer.

The sum of -47.03 and -22.71 is -69.74. Write this as the final solution: \(-47.03 + (-22.71) = -69.74\)

Key Concepts

Absolute Value

Understanding the concept of absolute value is crucial when working with negative numbers. Absolute value refers to the distance of a number from zero on a number line, regardless of its direction. Essentially, it is the non-negative value of a number without regard to its sign.

For instance, if you take the number \( -47.03 \), its absolute value is denoted as \( | -47.03 | = 47.03 \). This tells us how far the number is from zero, not considering whether it is to the left or right of zero on the number line. When adding negative numbers, as seen in the original exercise, the first step is often to look at the absolute values of the numbers in question.

• The absolute value of \( -47.03 \) is \( 47.03 \).
• The absolute value of \( -22.71 \) is \( 22.71 \).

After finding these values, you can add them together to find the magnitude of the answer before considering the sign.

Negative Numbers Arithmetic

When dealing with negative numbers arithmetic, it's important to keep track of the signs. Negative numbers are numbers that are less than zero, represented with a minus sign (e.g., \( -5 \)).

In arithmetic, whenever you add two negative numbers, the result will always be negative. This is because you are essentially moving further away from zero in the negative direction. To add such numbers:

• Find the absolute values and add them together.
• Since both numbers are negative, the result will also be negative.

Following the exercise we're discussing, we first ignore the negative signs and simply add the values: \( 47.03 + 22.71 = 69.74 \). Recognizing that both original numbers were negative, we then assign a negative sign to the result, ending up with \( -69.74 \).

Algebraic Addition

Algebraic addition involves combining numbers or variables to find a sum. When adding numbers algebraically, the key is to combine like terms and consider the signs of the numbers involved.

In the context of our exercise, \( -47.03 + (-22.71) \), we are adding together two negative numbers, which are like terms since they have the same sign. The rules of algebraic addition state that when you add two like terms with the same sign, you combine their absolute values and keep the common sign.

So, you'd:

• Add the absolute values: \( 47.03 + 22.71 = 69.74 \).
• Since both terms are negative, assign a negative sign to the sum: \( -69.74 \).

Therefore, the result of the algebraic addition is \( -69.74 \) as found in the exercise. This problem serves as a basic example of algebraic addition with negative numbers.