Problem 129 - Intermediate Algebra

Question

In the following exercises, translate to an algebraic expression and simplify if possible. (a) the difference of 10 and -18 (b) subtract 11 from -25

Step-by-Step Solution

Answer

(a) 28(b) -36

Step 1: Translate to Algebraic Expression (Part a)

The phrase 'the difference of 10 and -18' translates to the algebraic expression: 10 - (-18)

Step 2: Simplify the Expression (Part a)

Simplify the expression by applying the rule that subtracting a negative number is the same as adding a positive number: 10 - (-18) = 10 + 18

Step 3: Calculate the Simplified Expression (Part a)

Compute the sum: 10 + 18 = 28

Step 4: Translate to Algebraic Expression (Part b)

The phrase 'subtract 11 from -25' translates to the algebraic expression: -25 - 11

Step 5: Simplify and Calculate the Expression (Part b)

Compute the difference: -25 - 11 = -36

Key Concepts

algebraic expressions

Algebraic expressions are a way to represent mathematical phrases using numbers, variables, and operators. In our exercise, we translated verbal phrases into algebraic expressions. Let's explore these steps and concepts further.
For Part (a), the phrase 'the difference of 10 and -18' was converted into the algebraic expression: 10 - (-18).
For Part (b), the phrase 'subtract 11 from -25' became: -25 - 11.
Translating verbal phrases to algebraic expressions involves:

• Identifying keywords like 'difference', 'sum', 'product', 'quotient'.
• Recognizing whether the operation needed is addition, subtraction, multiplication, or division.
• Correctly placing numbers or variables in the expression based on their order in the phrase.

This skill is crucial in understanding and solving complex algebraic problems.

simplifying expressions

Simplifying expressions helps in making calculations easier and more understandable. When you simplify, you reduce an expression to its simplest form while keeping its value the same. Let's look at how this was applied in our exercise.
For Part (a), the expression 10 - (-18) was simplified by applying the rule: subtracting a negative number is like adding a positive. Thus, 10 - (-18) became 10 + 18.
Simplifying involves:

• Applying the correct rules for operations (like the one above).
• Combining like terms where possible.
• Breaking down complex expressions into manageable steps.

This ensures the problem becomes easier and quicker to solve. Learn and practice these rules to handle more challenging problems with confidence.

addition and subtraction of integers

Addition and subtraction of integers are fundamental arithmetic operations that are widely used in algebra. In our exercise, these operations were applied to solve the translated algebraic expressions.
For Part (a), once we converted the phrase to 10 - (-18), we changed it to 10 + 18 and then calculated the sum: 10 + 18 = 28.
For Part (b), the phrase translated to -25 - 11, and the difference was calculated as: -25 - 11 = -36.
Key points to remember in adding and subtracting integers:

• Adding a positive number moves you to the right on the number line.
• Adding a negative number (or subtracting a positive) moves you to the left.
• Subtracting a negative number is the same as adding its positive counterpart.

Practicing these operations strengthens your ability to tackle various mathematical problems with ease.