Problem 69 - Algebra for College Students

Question

Simplify each numerical expression. $$3(5-9)-3(-6)$$

Step-by-Step Solution

Answer

The simplified expression is 6.

Step 1: Simplify Inside the Parentheses

Start by addressing the operations within the parentheses. For the expression \(5 - 9\), this simplifies to \(-4\). Now, rewrite the expression as \(3(-4) - 3(-6)\).

Step 2: Multiply Coefficient with Parentheses Result

Next, multiply each number outside the parentheses with the result inside. For \(3(-4)\), multiply 3 by -4 to get \(-12\). For \(3(-6)\), multiply 3 by -6 to get \(-18\). This gives you the expression \(-12 - (-18)\).

Step 3: Simplify Double Negative

This step involves addressing the subtraction of a negative number, which is equivalent to addition. Therefore, \(-12 - (-18)\) simplifies to \(-12 + 18\).

Step 4: Calculate the Final Result

Finally, perform the addition. Add \(-12 + 18\) to get \(6\). Thus, the simplified expression is \(6\).

Key Concepts

Simplification

Simplification is a process that makes a mathematical expression easier to work with. It involves reducing an expression to its most basic form while maintaining its value. In our given problem, simplification is about reducing the expression \(3(5-9)-3(-6)\) to a single number.
This entails a sequence of steps such as resolving operations within parentheses, multiplying, and handling negative numbers. Each of these steps ensures that the final outcome is achieved in the simplest, most understandable form.
By breaking numerical expressions down, problems become easier to solve and understand, making it an indispensable part of algebra.

Parentheses

Parentheses play a crucial role in mathematical expressions. They indicate the order in which operations should be performed. When simplifying expressions like \(3(5-9)-3(-6)\), those inside the parentheses need to be solved first.
In our step-by-step solution, we first deal with the expression \(5 - 9\) inside the parentheses, simplifying it to \(-4\).
This changes our original expression to \(3(-4) - 3(-6)\). Addressing parentheses first ensures that calculations are performed correctly and in the proper sequence, aligning with the order of operations rules.

Negative Numbers

Negative numbers are numbers less than zero, and they require special attention in arithmetic operations. When working with negative numbers in expressions, the sign matters a lot.
For instance, in our expression, solving \(5 - 9\) gives us a negative number, \(-4\).
Similarly, there's the negative number \(-6\) in the term \(-3(-6)\). Knowing how to handle these correctly, by acknowledging that subtracting a negative is the same as adding its positive counterpart, is crucial. Therefore, \(-12 - (-18)\) translates to \(-12 + 18\).
Understanding these nuances helps to ensure algebraic precision.

Multiplication

Multiplication is another fundamental operation highlighted in this exercise. When you multiply a number outside the parentheses with the result inside, you simplify the expression further.
From our problem, multiplying \(3\) by \(-4\) gives \(-12\), and multiplying \(3\) by \(-6\) results in \(-18\).
This changes the task to handling the expression \(-12 - (-18)\). Multiplying sometimes involves working with negative numbers, applying the rule that a positive times a negative number is negative.
This operation, when executed correctly, ensures that the expression is reduced consistently and accurately, forming the basis for final calculations.