Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression for each value of \(x\). (If not possible, state the reason.)\(\begin{array}{lll}5-3 x & \text { (a) } x=-3 & \text { (b) } x=2\end{array}\)

Step-by-Step Solution

Answer

For \(x = -3\), the given expression evaluates to \(14\), and for \(x = 2\), the expression evaluates to \(-1\).

Step 1: Substitute \(x = -3\) in the given expression

The expression given is \(5 - 3x\). Substitute the value \(x = -3\) into the expression: it becomes \(5 - 3*(-3)\) which simplifies to \(5 + 9\). This is because subtracting a negative number would result in addition. So, the expression is evaluated as \(14\).

Step 2: Substitute \(x = 2\) in the given expression

Now, let's evaluate the expression for \(x = 2\). Substituting \(2\) for \(x\) in the given expression \(5 - 3x\), we get \(5 - 3*(2)\) which simplifies to \(5 - 6\). The evaluated expression thus gives the result \(-1\).

Key Concepts

Substitution Method

The substitution method is a straightforward strategy to evaluate expressions. It involves replacing the variable in an expression with a specified value. This method is crucial because it helps us determine the output of an expression for different inputs.
Using the substitution method for the expression given, such as in this exercise, we simply replace the variable \(x\) with a number. For instance:

• When \(x = -3\), the expression \(5 - 3x\) becomes \(5 - 3*(-3)\). Here, the substitution has been made by switching \(x\) with \(-3\).
• Similarly, substitute \(x = 2\) to get \(5 - 3*(2)\).

Simple as that! Remember: always do the multiplication before doing any addition or subtraction, following the order of operations known as BODMAS or PEMDAS. This ensures that the evaluation is accurate.

Simplification

Simplification is a powerful tool that allows us to make expressions more manageable and easier to understand. This process involves condensing expressions to their simplest form by performing arithmetic operations. The goal of simplification is to reduce complexity without changing the fundamental value.
For example, in our current problem, once we've substituted \(x = -3\) in \(5 - 3(-3)\):

• First, compute the multiplication \(-3 * (-3)\) which gives \(9\).
• Next, substitute the multiplication result into the expression and add to get \(5 + 9\).

The result is \(14\). The same steps apply for \(x = 2\), leading to \(5 - 6 = -1\).
Simplification removes unnecessary complexities and ultimately helps to clarify the mathematical expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations such as addition, subtraction, multiplication, and division. Understanding these expressions is fundamental in algebra, providing a way to represent real-world problems in mathematical form.
An algebraic expression can take many forms. In our case, \(5 - 3x\) is an example where:

• \(5\) is a constant, which remains fixed.
• \(3x\) involves a variable \(x\), which can change the expression's output value depending on \(x\)'s value.

Recognizing the components of an algebraic expression helps us manipulate and evaluate these expressions, such as substituting values and simplifying through basic arithmetic operations. This understanding serves as a foundation for solving equations and inequalities. By mastering these core concepts, evaluating expressions such as those in your exercise becomes a manageable task.