Problem 11 - Elementary Algebra

Question

Simplify. $$ 2+3(-2)-7 $$

Step-by-Step Solution

Answer

The simplified expression is -11.

Step 1: Distribute Multiplication

The expression given is: \[ 2 + 3(-2) - 7 \]Here, you need to carry out the multiplication as indicated by the parentheses. Multiply 3 by -2:\[ 3(-2) = -6 \]So the expression becomes:\[ 2 - 6 - 7 \]

Step 2: Simplify by Combining the Terms

Now, combine the terms in the expression:\[ 2 - 6 - 7 \]First, perform the subtraction with the first two numbers:\[ 2 - 6 = -4 \]Next, subtract 7 from -4:\[ -4 - 7 = -11 \]

Step 3: Final Result

The simplified expression is:\[ -11 \]

Key Concepts

Distributive Property

The distributive property is a valuable tool in algebra that helps you simplify expressions. It states that multiplying a single term by terms inside a parenthesis means multiplying the single term with each term inside. In simpler terms, when you see something like

• \( a(b + c) \)

You can use the distributive property to expand it to:

• \( ab + ac \)

In the exercise given, we applied this property by multiplying 3 by each term within the parenthesis \(-2\). Thus,

• \( 3(-2) = -6 \).

After distributing, the expression became simpler, leading us into the next step where numerical operations are involved. This makes complex expressions easier to manage and simplify.

Integer Operations

Integer operations involve the basic arithmetic processes you perform on whole numbers, both positive and negative ones. Mastering these operations is crucial for solving algebraic expressions accurately.
To handle integer operations:

• Add positive numbers like normal addition.
• When subtracting, think of it as adding the opposite (e.g., \(a - b = a + (-b)\)).
• Multiplying and dividing are straightforward: two positives make a positive, while two negatives also make a positive.
• A negative and a positive result in a negative.For instance, in the step \[ 2 - 6 - 7 \], we initially subtracted \( 6 \) from \( 2 \) to get \( -4 \). Then subtracted \( 7 \) from \( -4 \), which involved adding \(-7\) to \(-4\). This resulted in the final value of \(-11\). Understanding these rules helps in simplifying expressions correctly.

Order of Operations

The order of operations is an essential concept in mathematics to ensure calculations are carried out in the right sequence. Remember the phrase PEMDAS:

• Parentheses first
• Exponents next
• Multiplication and Division from left to right
• Addition and Subtraction from left to right

In our example, first we dealt with the parentheses by multiplying \(3\) by \(-2\) following the distributive property. No exponents were present, so we moved to the multiplication step. After that, we approached subtraction and addition from left to right as per PEMDAS.This systematic approach ensures that complex expressions are simplified accurately, avoiding the pitfalls of careless mistakes and providing consistent results every time.