Problem 75 - Algebra 2

Question

PREREQUISITE SKILL Evaluate each expression if \(a=2, b=-\frac{3}{4},\) and \(c=1.8 .(\text { lesson } 1-1)\) $$ -9(a-6) $$

Step-by-Step Solution

Answer

The expression evaluates to 36.

Step 1: Substitute the value of a

First, replace the variable \(a\) with its given value in the expression \(-9(a - 6)\). Since \(a = 2\), substitute \(2\) for \(a\) to get:\[-9(2 - 6)\]

Step 2: Solve inside the parentheses

Now solve the expression inside the parentheses. Subtract \(6\) from \(2\) to get:\[2 - 6 = -4\]So the expression becomes:\[-9(-4)\]

Step 3: Multiply the numbers

Finally, multiply \(-9\) by \(-4\). Since the product of two negative numbers is positive, you get:\[-9 \times -4 = 36\]

Key Concepts

Substitution

In algebra, substitution is a technique used to simplify expressions by replacing the variables with their assigned values. Substitution plays a central role when evaluating algebraic expressions. By replacing variables with numbers, we transform algebraic expressions into arithmetic ones. This helps simplify the solution process.
For instance, in the original exercise, we have the expression \(-9(a - 6)\). Here, the given value for \(a\) is 2. When we substitute \(2\) into the expression for \(a\), the expression becomes \(-9(2 - 6)\).
Substitution is straightforward but essential:

• Find the given value for each variable.
• Insert the values wherever they appear in the expression.
• Perform any further arithmetic operations needed.

Remember that substitution simplifies the expressions only up to a point. After substitution, further simplification involving arithmetic operations is usually necessary.

Simplification

Simplification is the process of breaking down a complex expression into a more manageable form. In our example, this consisted of calculating the result of the operation inside the parentheses, followed by multiplying the terms outside.
After substituting the given values, the next step is solving the operation inside the parentheses. In the expression \(-9(2 - 6)\), we first compute \((2 - 6)\):

• Perform the operation inside the parentheses: \(2 - 6 = -4\).

Simplification often involves:

• Performing arithmetic operations (addition, subtraction, multiplication, division).
• Reordering or rearranging terms to make expressions easier to manage.
• Combining like terms.

Once the expression within the parentheses has been simplified, you would handle any remaining operations, such as multiplying or dividing terms as needed. It's vital to simplify any problem to its core elements as much as possible, as this is critical in revealing the solution.

Negative Numbers

Negative numbers can initially seem tricky, but understanding them is essential for managing algebraic expressions smoothly. When dealing with negative numbers:

• If two negative numbers are multiplied or divided, the result is positive.
• If negative and positive numbers are multiplied or divided, the result is negative.

In our exercise, we handled the term \(-9(-4)\). Since both numbers are negative, multiplying them gives a positive outcome: \(-9 \times -4 = 36\). This follows the rule that the product of two negative numbers is always positive.
A handy mental note is to think of negative numbers as movements in the opposite direction. Thus, when you multiply two negative numbers, it's akin to taking two steps back, which effectively becomes a forward step, making the outcome positive.
With practice, working with negative numbers becomes second nature, and you'll be able to handle them effortlessly in more complex algebraic situations.