Problem 64 - Algebra for College Students

Question

Simplify each numerical expression. $$-9-4(-2)+(-7)(6)$$

Step-by-Step Solution

Answer

The simplified result is -43.

Step 1: Identify Groupings and Operations

First, examine the expression \[-9-4(-2)+(-7)(6)\]Look for parentheses and different operations in the expression to determine a simplifying order. You'll notice there is a multiplication within the parentheses and a subtraction/addition outside.

Step 2: Simplify Inside Parentheses

Focus on the term inside the parentheses, \(-4(-2)\). Multiplication has a higher precedence than addition/subtraction, so simplify \(-4(-2)\).This gives us a positive product: \(-4 \times -2 = 8\).

Step 3: Perform Remaining Multiplication

Next, solve the term \((-7)(6)\), which is the second multiplication operation within the expression. Multiply these numbers:\(-7 \times 6 = -42\).

Step 4: Combine Results

Substitute the simplified terms back into the expression:\(-9 + 8 - 42\).Now, perform the additions and subtractions from left to right. Start by adding:\(-9 + 8 = -1\),and then combine with the last term:\(-1 - 42 = -43\).

Step 5: Verify Final Result

Ensure that each simplification step followed the correct order of operations, which includes parentheses and multiplication before addressing addition and subtraction. After checking, confirm that the final simplified value for the expression is \(-43\).

Key Concepts

Order of Operations

When simplifying algebraic expressions, it’s crucial to follow the order of operations. This is like a set of rules that ensures everyone calculates expressions the same way. A common mnemonic to remember the order is PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and square roots, etc.)
• MD: Multiplication and Division (left-to-right)
• AS: Addition and Subtraction (left-to-right)

In our example, \[-9-4(-2)+(-7)(6),\]there are no exponents to worry about, but we do have parentheses and the basic operations of multiplication, addition, and subtraction. We start with the calculations inside the parentheses. Remember, if we were to ignore these rules, we could end up with a totally different, incorrect answer. By following PEMDAS, we're ensuring our solution is logical and consistent every time.

Multiplication and Subtraction

In algebra, getting comfortable with both multiplication and subtraction is essential. These operations are commonplace, and their order impacts the final solution significantly. Multiplication is performed before addition and subtraction unless governed by parentheses. In our expression, \[-9-4(-2)+(-7)(6),\]the process starts by solving the multiplication inside the parentheses: \(-4(-2)\). This results in a positive eight because a negative times a negative equals a positive.

• Next, handle the standalone multiplication: \((-7)(6),\)which equals \(-42.\)
• Finally, with these multiplications complete, proceed with subtraction and addition from left to right: \(-9 + 8 = -1\) followed by \(-1 - 42 = -43.\) The sequence in which subtraction and addition occur is determined after multiplication is resolved.

Paying attention to this order is necessary to avoid errors, leading to the expression's correct simplification.

Parentheses in Algebra

Parentheses play a pivotal role in algebra by altering the natural order of operations. They force you to address the enclosed calculations first. In our example, the parentheses \(-4(-2)\)indicate that the multiplication inside must be completed before any other operation takes place. This simplifies to positive \(8,\)given that multiplying two negative numbers yields a positive value.

Once you remove the expression inside the parentheses, the multiplication \((-7)(6)\)can be tackled next since there are no further markers of priority over multiplication.

By clarifying which operations to perform first, parentheses ensure your algebraic solutions follow a logical sequence. Handling expressions correctly directly hinges on first resolving these grouped calculations. So, whenever you see parentheses, remember: your first stop in order management!