Problem 136 - Introductory Algebra for College Students

Question

The list shows a pattern for various products. $$\begin{aligned}2(-3) &=-6 \\\1(-3) &=-3 \\\0(-3) &=0 \\\\-1(-3) &=3 \\\\-2(-3) &=6 \\\\-3(-3) &=9 \\\\-4(-3) &=? \end{aligned}$$ Use this pattern to find \(-4(-3)\)

Step-by-Step Solution

Answer

The product of -4 and -3 following the pattern is 12.

Step 1: Understand the Pattern

Look carefully at the list and identify the pattern. It can be seen that in each case, multiplying a negative number by -3 makes the result positive and increments it by 3 each time.

Step 2: Apply the Pattern

Continue the pattern by taking the last established result (which was 9 for -3*(-3)), and increment it by 3 because that's the pattern discovered, to find the product of -4 and -3. Thus, the result becomes 9+3=12.

Key Concepts

Multiplication Patterns

Understanding multiplication patterns can be incredibly useful, especially in algebra. Patterns are consistent ways that numbers behave under different operations, and by identifying them, you can solve problems more efficiently.

In this particular exercise, the pattern involves multiplying a series of descending integers (starting with positive and moving to negative) by \(-3\). Here's what the pattern reveals:

• When multiplying any positive integer by a negative integer, the result is negative.
• As the integers decrease by 1, the product increases by 3 in absolute value.
• When you reach a negative integer multiplied by another negative, like \(-1(-3)\), the result turns positive.

This pattern continues as negative integers are further multiplied by \(-3\). The absolute value of the product increases by 3 each time. Recognizing this pattern helps us find products quickly without recalculating from scratch. In our exercise, the pattern was consistent and let us predict the answer efficiently.

Negative Numbers

Negative numbers can initially seem tricky, but with a bit of practice, they become easy to manage. In real life, negative numbers often represent concepts like loss or debt. In mathematical terms, they're values below zero.

When working with negative numbers in multiplication, there are a few general rules to remember:

• Multiplying a positive number by a negative number yields a negative result.
• The product of two negative numbers is positive.
• Multiplying any number by zero results in zero.

In the given exercise, these rules are at play. For example, multiplying by \(-3\) repeatedly switches the sign as the original number changes from positive to negative. This results in a pattern where the outcome flips from negative to positive as explained. Remembering these rules helps simplify calculations and solve algebraic problems more easily.

Mathematical Reasoning

Mathematical reasoning involves using logical thinking to solve problems. It is critical in algebra where patterns and equations can be solved step by step by following logical processes.

In the exercise, reason was applied both to identify the pattern and continue it logically to find the answer to \(-4(-3)\).

• First, recognize the pattern in numbers and their increments, both in their positive or negative results.
• Next, establish what the pattern dictates as inputs are continued, moving from \(-3 \times -3 = 9\) to another step by adding 3.
• The last logical step is applying this understanding to new or unknown parts of a sequence to predict results accurately.

This systematic approach ensures you draw the correct conclusions, making complex ideas more manageable. With practice, your mathematical reasoning skills improve, allowing you to see solutions clearly and logically.