Problem 35 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 11-(-23)-77 $$

Step-by-Step Solution

Answer

The value of the given expression is -43.

Step 1: Understanding subtraction of a negative number

Subtracting a negative number is equivalent to addition. So, the expression \(11-(-23)-77\) can be rewritten as \(11+23-77\).

Step 2: Perform addition

Now, add the first two numbers in the expression: \(11+23 = 34\). So, the expression becomes \(34 - 77\).

Step 3: Perform subtraction

Finally, subtract 77 from 34. This will give you a negative result since 77 is larger than 34. The operation \(34 - 77 = -43\).

Key Concepts

Subtraction

Subtraction is one of the fundamental operations in mathematics. It involves taking away a quantity from another. When we subtract, we essentially determine how much is left when a part is removed from the whole. In the expression given, subtraction is seen twice: first dealing with a negative number and then as a direct operation. Here’s a deeper look at how subtraction functions:

• Identify the numbers involved in the problem. Here, you have numbers like 11 and -23.
• Understand how they interact. In an equation like this, notice that subtraction can often lead to unexpected results due to signs of the numbers involved.
• Always make room for subtraction leading to negative results, especially when a larger number is subtracted from a smaller one.

Subtraction doesn't just take numbers away; it opens up a pathway to understanding other concepts, especially in algebra.

Negative Numbers

Negative numbers are numbers less than zero. They are integral to arithmetic and algebra, providing a simple way to express quantities less than nothing. Their understanding is crucial in evaluating expressions like the one in this exercise. When you come across a negative number in a subtraction problem, something special happens:

• Subtracting a negative is the same as adding a positive. This arises because two negatives make a positive.
• If you see a minus sign before a negative number, as in \(11 - (-23)\), it changes to addition: \(11 + 23\).

Negative numbers help in representing temperatures below zero, debts, and many physical quantities in the opposite direction of a defined positive. Properly handling them ensures your solutions remain accurate.

Addition

Addition is the process of combining two or more numbers to get a total. In the context of this exercise, addition comes into play after transforming the subtraction of a negative number into addition. Here is how addition simplifies our expression:

• By changing \(11 - (-23)\) to \(11 + 23\), we simplify the calculation.
• Addition combines \(11\) and \(23\) to make \(34\), simplifying our expression considerably.

Addition helps accumulate values to reach a total. It’s foundational to understanding more complex operations. Remember, addition is both commutative and associative:

• Commutative means the order doesn’t matter: \(a + b = b + a\).
• Associative means you can group numbers differently without affecting the result: \( (a + b) + c = a + (b + c)\).

Understanding addition alongside subtraction and negative numbers arms you with the skills to tackle various algebraic expressions.