Problem 23 - Elementary and Intermediate Algebra

Question

Evaluate using a calculator. $$ 27 a-18 b, \text { for } a=136 \text { and } b=13 $$

Step-by-Step Solution

Answer

3438

Step 1: Substitute the variables with given values

Replace \(a\) and \(b\) with the given values \(136\) and \(13\) respectively. The expression becomes: \[ 27 \times 136 - 18 \times 13 \]

Step 2: Perform the multiplications

First, calculate \(27 \times 136\). Using a calculator, \[27 \times 136 = 3672\]. Next, calculate \(18 \times 13\). Using a calculator, \[18 \times 13 = 234\].

Step 3: Subtract the products

Subtract the second product from the first product: \[3672 - 234 = 3438\]

Key Concepts

substitution method

The substitution method is a technique used in algebra to replace variables with their given values. For example, if you have the expression \(27a - 18b\) and you are given that \(a = 136\) and \(b = 13\), you would substitute \(a\) and \(b\) with these values. This process transforms the variable-based expression into a numerical one. Substitution simplifies solving and helps in breaking down more complex algebraic problems.
Follow these simple steps when using the substitution method:

• Identify the variables that need to be replaced.
• Insert the given numerical values in place of these variables.
• Proceed with arithmetic operations as usual.

Substitution is a fundamental step, especially for solving equations and simplifying expressions. It is used widely in both basic and advanced algebra.

arithmetic operations

Once you have substituted the variables in the expression, the next step involves performing arithmetic operations such as multiplication, addition, or subtraction. In our example:
After substituting, the expression becomes: \[ 27 \times 136 - 18 \times 13 \] Let's break down these operations:

• First, calculate the multiplication: \[ 27 \times 136 = 3672 \] Using a calculator makes this step easier and faster.
• Next, perform the second multiplication: \[ 18 \times 13 = 234 \]
• Finally, subtract the second product from the first: \[ 3672 - 234 = 3438 \]

These steps illustrate basic arithmetic operations that are essential for algebraic solutions. It's important to follow the sequence of operations carefully to arrive at the correct result. Familiarity with these operations ensures accuracy when solving algebraic equations.

using a calculator

Using a calculator can significantly simplify your calculations, especially for large numbers. Here’s how to use a calculator effectively in algebraic evaluation:

• Enter the multiplication expressions one at a time. For instance, for \(27 \times 136\), type in 27, press the multiplication sign, then 136, and press equals.
• Note down the result, which is 3672 in our example.
• Clear the calculator, then enter the next expression: \(18 \times 13\).
• Record this result as well, which is 234.
• Finally, subtract the second result from the first: Enter 3672, press the subtraction sign, type in 234, and press equals.

Using a calculator helps in avoiding manual calculation mistakes and saves time. It is a handy tool for both simple and complex arithmetic operations. The key to success is ensuring each step is performed accurately and sequentially.