Problem 22 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ -36-51 $$

Step-by-Step Solution

Answer

-87

Step 1: Rewrite the expression

To solve the subtraction problem, start by rewriting the expression \(-36 - 51\). Notice that subtracting a positive number is equivalent to adding a negative. So, \(-36 - 51\) can be rewritten as \(-36 + (-51)\).

Step 2: Add the numbers

Now, add the two negative numbers \(-36 ext{ and }-51\). When adding two negative numbers, you add their absolute values and keep the negative sign. Calculate the sum: \(36 + 51 = 87\).

Step 3: Apply the negative sign to the result

Since both numbers were negative, the result of the addition is also negative. Therefore, the final answer is \(-87\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero, expressed with a minus sign (-). They represent quantities below a neutral point, often used to indicate losses, deficiencies, or values in reverse direction. Negative numbers are tricky in arithmetic operations because they follow different rules compared to positive ones.

• Negative times Negative gives a Positive.
• Negative times Positive gives a Negative.
• Adding two negative numbers means you add their absolute values and put a negative sign in front of the result.

For the given problem, understanding this helps in rewriting subtraction as addition. For instance, subtracting a positive can be rewritten as adding its negative equivalent.

Working with Algebraic Expressions

Algebraic expressions involve variables, numbers, and operations. They form equations and phrases that express mathematical relationships. In these expressions, subtraction can often be transformed to addition, using negative numbers.

When dealing with algebraic expressions:

• Recognize parts of the expression. Identify terms, coefficients, and variables.
• Simplify expressions by combining like terms.
• Transform expressions as needed for easier calculations, especially with subtraction.

In our example, \(-36 - 51\) was made simpler by recognizing it as \(-36 + (-51)\). This transformation aids in understanding and computation.

Integer Operations Simplified

Integer operations include addition, subtraction, multiplication, and division of whole numbers, either positive or negative. Mastering these is crucial to solving algebraic problems effectively.

For addition and subtraction of integers, keep the following in mind:

• Addition: When signs are the same (both positive or both negative), add the absolute values and keep the common sign.
• Subtraction: Change the subtraction sign to addition and reverse the sign of the subsequent number.

In the subtraction \(-36 - 51\), a concept is converting it into an addition problem \(-36 + (-51)\), which involves calculating the sum of their absolute values and keeping the negative sign, resulting in the answer \(-87\). This simplification aids in efficient problem-solving with integers.