Problem 43 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(-3(-a)(-a)\) when \(a=-7\)

Step-by-Step Solution

Answer

-147

Step 1: Substitute the Given Value

The problem already provides the value of the variable 'a' which is -7. Now substitute this value into the expression. So, the expression will become: -3(-(-7))(-(-7))

Step 2: Evaluate the Inner Parentheses

Next, simplify the expression inside the parentheses. According to the rules of negation, negating a negative number results in a positive number. Therefore, -(-7) simplifies to 7. Now, replace -(-7) with 7 in the original expression. So, the expression becomes: -3(7)(7)

Step 3: Perform the Multiplication

Now perform the multiplication from left to right: -3 times 7 equals -21 and -21 times 7 equals -147.

Key Concepts

Substitution in Algebra

One of the cornerstone techniques in algebra is the process of substitution. This process involves replacing a variable in an expression with its corresponding numerical value. Let's explore how substitution works with a clear example.

Consider you have an expression like \( -3(-a)(-a) \) and you are told that \( a = -7 \). The 'substitution' here means that every instance where 'a' appears in the expression, you replace it with \( -7 \). It's important to do this carefully to avoid any sign errors.

After replacing the variable 'a' with \( -7 \) in the above expression, you are actually simplifying the equation step by step – which brings us to our next important concept.

Simplifying Expressions

Once the substitution is done, the next step in evaluating an algebraic expression is simplifying it. Simplification involves performing operations and reducing the expression to its simplest form. For example, after substituting \( a = -7 \) into the expression \( -3(-a)(-a) \), you get \( -3(-(-7))(-(-7)) \).

Simplification starts with handling the parentheses. A negative multiplied by a negative gives a positive, thus \( -(-7) \) becomes \( 7 \). The result is \( -3(7)(7) \), a much simpler expression. Simplifying expressions requires attention to order of operations and understanding that multiplication and division come before addition and subtraction. It's always crucial to do these steps one at a time to avoid mistakes.

Through simplification, the expression 'shrinks' in complexity, making it easier to understand and solve, which seamlessly transitions into the multiplication of the simplified terms.

Multiplication of Integers

Understanding the rules for the multiplication of integers is essential in algebra. This includes knowing that multiplying two negatives or two positives results in a positive, while a positive and a negative give a negative. In our original expression, we are left with \( -3(7)(7) \).

We then proceed to multiply these integers. Start with \( -3 \) and \( 7 \) — since one is negative and the other is positive, the result is negative: \( -3 \times \( 7 \) = -21 \). Now multiply \( -21 \) and \( 7 \) which are also of different signs, thus the result remains negative: \( -21 \times \( 7 \) = -147 \).

Remember, taking it step by step prevents errors and ensures that each multiplication is handled correctly. Multiplication is commutative and associative, but it is always safer to follow the sequence of the given operations, especially when working with negative numbers.