Problem 1 - Basic Engineering Mathematics

Question

Evaluate \(3 a b-2 b c+a b c\) when \(a=1, b=3\) and \(c=5\)

Step-by-Step Solution

Answer

The evaluated result is -6.

Step 1: Substitute Values

First, substitute the given values of variables into the expression. Here, substitute \(a=1\), \(b=3\), and \(c=5\) into the expression \(3ab - 2bc + abc\). This results in the expression: \(3(1)(3) - 2(3)(5) + (1)(3)(5)\).

Step 2: Calculate Each Term Separately

Next, calculate each term individually using the substituted values. For the first term, calculate \(3(1)(3) = 9\). For the second term, calculate \(2(3)(5) = 30\). For the third term, calculate \((1)(3)(5) = 15\).

Step 3: Combine the Calculations

Now, combine the calculated values from each term. The expression becomes \(9 - 30 + 15\).

Step 4: Perform the Final Calculation

Finally, perform the arithmetic operations: \(9 - 30 + 15\). First, subtract 30 from 9 which results in \(-21\). Then add 15 to \(-21\) to get \(-6\).

Key Concepts

Understanding the Substitution Method

The substitution method is a crucial tool in algebra that helps simplify expressions by replacing variables with actual numerical values. In our exercise, we have an algebraic expression, \(3ab - 2bc + abc\), with specific values for each variable: \(a=1\), \(b=3\), and \(c=5\).
The main idea is to take each instance of the variable in the formula and replace it with its given value.
This process transforms an otherwise abstract expression into something more tangible that we can compute step-by-step. Here, the substitution gives us:\(3(1)(3) - 2(3)(5) + (1)(3)(5)\).

• Ensure you replace every occurrence of the variable to avoid calculation errors.
• Write the expression clearly after substitution to see numerical operations that follow.

Once you have substituted the numbers, you're ready to perform calculations without going back to the original variables again.

Mastering Arithmetic Operations

Once you have substituted the values into an algebraic expression, the next step involves performing arithmetic operations. Arithmetic operations are fundamental processes including addition, subtraction, multiplication, and division.
To solve our given expression, it is crucial to manage these operations systematically. Let's break down the steps for our expression with the inserted values:

First, calculate each term one by one:

• \(3(1)(3)\): Perform multiplication first, yielding \(9\).
• \(-2(3)(5)\): This becomes \(-30\) as you compute \(-2\cdot3\cdot5\).
• \((1)(3)(5)\): This results in \(15\).

After calculating each separate entity, combine them back together appropriately:

Now, address them as \(9 - 30 + 15\). Each operation follows the standard path of arithmetic: subtraction first and then addition. Ensure you follow the order of operations (PEMDAS/BODMAS rules), as they dictate calculation precedence.

Evaluating Variables Correctly

Evaluating variables means substituting them with real numbers and computing the final expression result. It's important to understand the role each variable plays in an expression and to assess correct outcomes. In our example, the variables \(a\), \(b\), and \(c\) were evaluated into the expression through substitution.

You should:

• Check your initial values to confirm accurate representation within the expression.
• Understand each variable’s contribution to the term it’s part of.
• After substitution changes the expression into numbers, ensure all resultant arithmetic operations are precise.

In our case, the expression transformed into numbers, was simplified through arithmetic operations and evaluated as follows: Calculate \(3ab - 2bc + abc\) given values \(a=1\), \(b=3\), and \(c=5\), the result was \(-6\).
Remember, careful evaluation of each variable's impact ensures clarity and correctness, leading to successfully solving algebraic problems.