Problem 39 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$\frac{6(-7)+3(-2)}{20-4}$$

Step-by-Step Solution

Answer

The simplified expression is -3.

Step 1: Simplify the Numerator

First, simplify the numerator by applying the distributive property within the expression \(6(-7) + 3(-2)\). Calculate each multiplication separately: \(6(-7) = -42\) and \(3(-2) = -6\). Then, add these results together: \(-42 + (-6) = -48\). So, the simplified numerator is \(-48\).

Step 2: Simplify the Denominator

Next, simplify the denominator \(20 - 4\). Subtract \(4\) from \(20\) to get \(16\).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now, divide the simplified numerator \(-48\) by the simplified denominator \(16\). This gives \(-48 \div 16 = -3\).

Step 4: Final Step: Write the Simplified Expression

The final simplified expression is \(-3\).

Key Concepts

Order of Operations

When simplifying mathematical expressions, it's crucial to follow the order of operations. This ensures that everyone solves an expression the same way and arrives at the correct answer. A common acronym to help remember this order is PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

Start with operations inside the parentheses and work your way through exponents, then multiplication and division from left to right, followed by addition and subtraction also from left to right. This pecking order prevents mistakes, like performing addition before dealing with multiplication. For our exercise, each step respected this sequence, leading us neatly to the correct simplified expression.

Distributive Property

The distributive property is a valuable tool in algebra, especially when simplifying expressions. It says that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results. So, for the expression \( a(b + c) \), it is the same as \( ab + ac \). This property helps us to untangle expressions where you're multiplying a sum or difference.

In our exercise, we used the distributive property to simplify the expression \(6(-7) + 3(-2)\). By applying the distributive property, we multiplied 6 by -7 and 3 by -2 separately: \(6 \times -7 = -42\) and \(3 \times -2 = -6\). Adding the results gave us the new simplified expression in the numerator, which was -48.

Multiplication and Division

Multiplication and division are fundamental operations often performed before addition and subtraction when simplifying expressions. It's essential to remember that these operations should be done as they appear from left to right, following the order of operations.

In the given problem, multiplication was performed first within the numerator using the distributive property, resulting in \(-42\) and \(-6\). Once these results were obtained, addition was performed to combine these into \(-48\) before dividing by the simplified denominator.

The division step involved taking the simplified numerator \(-48\) and dividing it by 16, simplifying the expression further to \(-3\). Each step required careful consideration of order, ensuring that multiplication was addressed before moving to the final division.

Addition and Subtraction

Addition and subtraction are typically the last operations when simplifying expressions according to the order of operations. They should be carried out from left to right after all multiplication and division tasks are completed.

In our original exercise, after applying the distributive property and simplifying the multiplication, we performed addition in the numerator, which consisted of adding \(-42\) to \(-6\), resulting in \(-48\). This operation prepared our expression for the subsequent division by the denominator, following the order of operations precisely.

Understanding the proper placement for addition and subtraction in the hierarchy of operations is key to reaching a correct solution. It ensures that earlier multiplications or divisions don't lead to incorrect simplifications of an expression.