Problem 35 - Fundamentals of Mathematics

Question

Find the value of each expression. $$3(9 a-6 b), \text { if } a=-1 \text { and } b=-2$$

Step-by-Step Solution

Answer

The value of the expression is 9.

Step 1: Substitute the values of a and b

Given the expression \(3(9a - 6b)\), we need to substitute \(a = -1\) and \(b = -2\) into the expression. This gives us \(3(9(-1) - 6(-2))\).

Step 2: Simplify Inside the Parentheses

First, calculate \(9(-1)\) which equals \(-9\). Then, calculate \(-6(-2)\) which equals \(+12\). Therefore, the expression inside the parentheses simplifies to \(-9 + 12\).

Step 3: Simplify the Expression Inside the Parentheses

Simplify \(-9 + 12\) to get \(3\).

Step 4: Multiply by the Constant Outside

Now multiply the simplified result inside the parentheses by the constant outside: \(3 \times 3\).

Step 5: Compute the Final Result

Calculate \(3 \times 3\) to get \(9\).

Key Concepts

Substitution Method

The substitution method is a powerful tool in algebra, helping us to quickly evaluate expressions by replacing variables with given values. This technique streamlines solving efforts and reduces complex problems into simple arithmetic.
To use the substitution method, follow these steps:

• Identify the variables in your algebraic expression.
• Plug in the given values for these variables.

In our example, the expression is \(3(9a - 6b)\).
You're instructed to replace \(a\) with \(-1\) and \(b\) with \(-2\).
After substitution, the expression becomes \(3(9(-1) - 6(-2))\).
This method simplifies the initial problem into straightforward calculations.
When done correctly, substitution ensures you're working with accurate figures in subsequent steps.

Simplifying Expressions

Once you've substituted the values in your expression, the next step is simplifying it. Simplifying involves reducing the expression to its most basic form, which makes calculations more accessible.
Simplification typically involves:

• Combining like terms.
• Performing arithmetic operations.

Let's take a look at our expression from the example: \(3(9(-1) - 6(-2))\).
First, calculate \(9(-1)\), resulting in \(-9\), and \(-6(-2)\), which equals \(+12\).
Now, you'll have to simplify \(-9 + 12\).
This operation results in \(3\).
Reducing the expression in parentheses first helps in managing complex calculations easily.

Order of Operations

The order of operations is a fundamental principle that dictates the sequence in which operations should be performed to ensure consistent results. The acronym PEMDAS is often used as a mnemonic device to remember the order:

• P: Parentheses
• E: Exponents
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

In our expression, \(3(9(-1) - 6(-2))\), start with the operations inside the parentheses:
Calculate multiplication followed by subtraction, reducing it to \(3\).
Next, proceed with multiplication outside the parentheses: \(3 \times 3\).
This ensures that every operation is performed in the proper sequence, resulting in the correct final answer of \(9\).
Understanding and applying the order of operations is crucial for handling any mathematical expression accurately.