Problem 48 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$2(3-7)+3(5-6)$$

Step-by-Step Solution

Answer

The simplified expression is -11.

Step 1: Simplify Inside the Parentheses

The expression contains two sets of parentheses. Start by evaluating the expressions within each set of parentheses: \[3 - 7 = -4\] and \[5 - 6 = -1\]. Substitute back into the expression to get: \[2(-4) + 3(-1)\].

Step 2: Perform Multiplication

Now, perform the multiplication for each term: \[2(-4) = -8\] and \[3(-1) = -3\]. This simplifies our expression to: \[-8 + (-3)\].

Step 3: Perform Addition/Subtraction

Finally, evaluate the remaining expression by performing the addition/subtraction:\[-8 + (-3) = -11\].

Key Concepts

Addition and Subtraction

Addition and subtraction are the fundamental building blocks of arithmetic. They involve combining or removing numbers to find a result. When working with addition and subtraction in mathematical expressions, it’s important to perform these operations in the correct sequence, especially when they are mixed together.
To determine the right order, we follow the rule of operations known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). According to this rule, addition and subtraction should be done from left to right, but after any operations included in parentheses and any multiplication or division have been completed in the sequence.
In our example, after handling the multiplication, we are left with -8 + (-3). Here, we are effectively adding a negative number, which is the same as subtracting that number. Therefore:

• Think of -8 + (-3) as -8 - 3.
• This results in -11 because you are moving 3 units further negative from -8.

This might seem tricky at first, but practice makes perfect. By keeping track of signs and remembering the sequence of operations, addition and subtraction become less challenging.

Multiplication

Multiplication is a powerful arithmetic operation used to calculate the product of two numbers. It can simplify repeated addition, making operations more efficient and manageable.
In mathematics, multiplication can appear straightforward until we encounter negative numbers. A key rule is that multiplying two negative numbers results in a positive, while a positive multiplied by a negative results in a negative number. This is crucial when simplifying expressions.
In our problem: -4 and -1 are results from subtraction inside the parentheses.
Let’s break it down:

• First, calculate 2(-4): It means we have two times -4, which equals -8 because a positive times a negative yields a negative product.
• Next, evaluate 3(-1): Here, three times -1 equals -3 for the same reason.

After completing these multiplications, we're left with simple addition/subtraction, which leads us seamlessly to our final answer. Getting used to these multiplication principles, especially with negative numbers, will enhance your problem-solving skills in algebraic expressions.

Parentheses in Math

Parentheses are used in math to dictate the order in which certain operations should be performed. They are crucial for organizing calculations and can completely change the outcome of an expression if misused.
When you see an expression with parentheses, your first task is to simplify what’s inside. This takes precedence over any other operation outside of them. It’s like focusing on one part of a puzzle before tackling the larger picture.
In our example, before any other operations, we simplified within the parentheses:

• First, we solved 3 - 7 to get -4.
• Then, we handled 5 - 6 to obtain -1.

Replacing these into the main equation gave us 2(-4) + 3(-1), which was then ready for further calculations. This step ensures precision in mathematical expressions and prevents any mix-up of operations. Remember, without addressing parentheses first, the final result can easily end up incorrect.