Problem 129 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. \(a-b-c,\) for \(a=4, b=-2,\) and \(c=9\)

Step-by-Step Solution

Answer

\(-3\)

Step 1: Substitute given values into the expression

Replace each variable in the expression \(a-b-c\) with their respective values. Substituting \(a = 4,\, b = -2,\, c = 9,\) we get \(4 - (-2) - 9\).

Step 2: Simplify the expression

Solving the expression, remember that subtraction of a negative number becomes addition. So, the expression will change to \(4 + 2 - 9\). Simplifying further, we get \(6 - 9\).

Step 3: Calculate the final answer

Finally, we subtract 9 from 6 to get the final answer, which is \(-3\).

Key Concepts

Substituting Variables

When we talk about substituting variables in algebra, we are referring to the act of replacing the letters (or variables) in an expression with the values they represent. This is an essential first step in the process of evaluating expressions.

For example, in the given exercise, the variables are replaced with their respective numerical values: the variable 'a' is substituted with 4, 'b' with -2, and 'c' with 9. Substitution should be done with care, ensuring that you keep track of negative signs and any other operations that are part of the expression.

Common Substitution Pitfalls

• Forgetting to replace all instances of the variable.
• Overlooking negative signs, which can change the outcome drastically.
• Ignoring the operational context of the variable, such as if it’s part of an exponent or under a radical.

Remember that substitution is just the starting point to finding the value of an expression.

Simplifying Expressions

Once the variables in an algebraic expression have been substituted with actual values, the next step is simplifying the expression. This involves performing the arithmetic operations in the correct order and combining like terms if necessary.

Simplifying might also involve recognizing that the subtraction of a negative value is the same as addition – a common point of confusion that can trip up many students. In our example, after substitution, we simplify the expression by first addressing the double negative: goes from \(4 - (-2) - 9\) to \(4 + 2 - 9\). Afterward, we combine what we can, which in this case are the numbers 4 and 2, leading to further simplification.

Things to Remember While Simplifying

• Follow the order of operations (PEMDAS/BODMAS).
• Combine like terms correctly.
• Be mindful of negative signs and other operations.

The end goal is to make the expression as straightforward as possible, often resulting in a single number.

Arithmetic Operations

After simplifying the expression with the substituted values, the last hurdle is executing the arithmetic operations to find the final result. This usually involves adding, subtracting, multiplying, or dividing numbers to arrive at a solution.

In the provided exercise, after simplifying, we carry out the arithmetic by subtracting 9 from 6, which gives us the final answer of gives \(-3\). This might seem straightforward, but errors in this stage can undo the work done in the earlier steps.

Ensuring Accurate Arithmetic

• Double-check your work to catch any simple arithmetic errors.
• If you are combining several operations, prioritize them according to the rules (again, PEMDAS/BODMAS).
• If the numbers are large or complex, consider writing down intermediary steps to avoid confusion.

Attention to detail in arithmetic operations can make the difference between the correct and incorrect final answers.