Problem 47 - Algebra 1

Question

Simplify the expression. $$-7 \cdot\left(-\frac{2 w}{-7}\right)$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-2w\).

Step 1: Observe the expression.

The given expression is \(-7 \cdot\left(-\frac{2 w}{-7}\right)\). In order to simplify, the term within the parentheses should first be simplified.

Step 2: Simplify the term inside parentheses.

Inside the parentheses, we have \(-\frac{2 w}{-7}\). A negative divided by a negative yields a positive. Therefore, this simplifies to \(\frac{2 w}{7}\).

Step 3: Multiply the result from Step 2 by \(-7\).

Now we have \(-7 \cdot \frac{2 w}{7}\). Multiply \(-7\) by \(\frac{2w}{7}\) to get \(-2w\).

Key Concepts

Understanding Negative Numbers

Negative numbers can be confusing, but they aren't too challenging once you get the hang of them. A negative number is simply a number that is less than zero. Think of it as counting backwards from zero.
Negative signs (-) are used in math to indicate that a number is negative. When dealing with negative numbers in calculations:

• A negative plus a negative equals a negative.
• A positive plus a positive equals a positive.
• A negative added to a positive will depend on which is larger; subtract the numbers and take the sign of the larger.
• When you multiply or divide two negative numbers, the result is positive.
• When you multiply or divide a negative number by a positive number, the result is negative.

In the exercise, we observe two negative numbers being divided in the term \(-\frac{2w}{-7}\) which results in a positive \(\frac{2w}{7}\). Understanding these rules helps demystify operations involving negatives.

Mastering Multiplication with Fractions

Multiplying with fractions might initially seem tricky, but it's quite straightforward. The key is to multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
For instance, when multiplying a whole number by a fraction, you treat the whole number as a fraction with a denominator of 1.
Consider the example from the exercise, multiplying \(-7\) by \(\frac{2w}{7}\):

• First, represent \(-7\) as \(\frac{-7}{1}\).
• Next, multiply the numerators: \(-7 \times 2w = -14w\).
• Then, multiply the denominators: \(1 \times 7 = 7\).
• Thus, the result is \(\frac{-14w}{7}\), which simplifies further in the process.

Remember that it's important to always simplify fractions whenever possible.

Simplifying Expressions

Simplifying expressions entails reducing them into their simplest form to make calculations easier. It often involves combining like terms or reducing fractions.
In this exercise, we began with the expression \(-7 \cdot \left(-\frac{2w}{-7}\right)\):

• First, we simplified the fraction within the parentheses, turning \(-\frac{2w}{-7}\) into \(\frac{2w}{7}\) by eliminating the negative signs.
• Second, we multiplied \(-7\) by \(\frac{2w}{7}\).
• This multiplication gave \(\frac{-14w}{7}\), which we then simplified to \(-2w\) by dividing the numerator by the denominator.

The goal of simplification is to make expressions as manageable and clear as possible.