Problem 68 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ -5(6)(a) \text { when } a=-2 $$

Step-by-Step Solution

Answer

The value of the expression for \( a=-2 \) is 60.

Step 1: Substitute the known value

Replace the variable \( a \) in the expression with the given value, -2. It gives: -5 * 6 * -2

Step 2: Apply multiplication

Perform the multiplication operation. Since multiplication is associative, you can multiply in any order. Let's do it from left to right, first multiply -5 and 6, yielding -30, then multiply -30 with -2.

Step 3: Simplify

Simplify the expression obtained in step 2. The multiplication -30 * -2 yields 60.

Key Concepts

Substitution in Expressions

In mathematics, expressions often contain variables representing unknown or changing values. To evaluate an expression, we substitute these variables with known values. This process is called substitution.
For example, consider the expression \(-5 \times 6 \times a\). If we know that \(a = -2\), we substitute \(-2\) in place of \(a\) in the expression. This substitution transforms the expression into \(-5 \times 6 \times -2\).
By substituting correctly, we can turn abstract expressions into numerical ones, making them easier to solve. Substitution is a powerful tool in algebra and is commonly used in solving equations.

Multiplication of Integers

Multiplying integers follows a simple set of rules, especially regarding the signs:

• A positive number times a positive number gives a positive result.
• A negative number times a negative number also gives a positive result.
• A positive number times a negative number results in a negative number.

In our example, we have the expression \(-5 \times 6 \times -2\).
Let's break it down:

• First, multiply \(-5\) by \(6\) to get \(-30\). Since one is negative and the other is positive, the result is negative.
• Next, multiply \(-30\) by \(-2\). Both numbers are negative, so the product is positive and equals \(60\).

By keeping these rules in mind, we can efficiently multiply any integers regardless of their signs.

Associative Property of Multiplication

The associative property of multiplication states that the way in which numbers are grouped in a multiplication problem does not affect the product. This means you can multiply numbers in any order and achieve the same result. The formula for this property is \((a \times b) \times c = a \times (b \times c)\).
Let's apply this property to our expression \(-5 \times 6 \times -2\): - You can choose to first multiply \(-5\) and \(6\), and then multiply the result \(-30\) by \(-2\), which gives us \(60\). - Alternatively, you could multiply \(6\) by \(-2\) first to get \(-12\), and then multiply \(-5\) by \(-12\), also resulting in \(60\).
This flexibility in multiplication order can sometimes simplify calculations, allowing you to approach problems in the most convenient way.