Problem 14 - Beginning and Intermediate Algebra

Question

Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(7 x-9\)

Step-by-Step Solution

Answer

For (a) x=4: 19; For (b) x=6: 33

Step 1: Substitute for part (a)

Substitute the value of 4 for the variable x in the given expression: The given expression is: Substitute x = 4: \(7(4) - 9\)

Step 2: Simplify the expression for part (a)

Calculate the value inside the parentheses first: \(7 \cdot 4 = 28\)Then subtract 9 from 28: \(28 - 9 = 19\)

Step 3: Substitute for part (b)

Substitute the value of 6 for the variable x in the given expression: The given expression is: Substituting x = 6: \(7(6) - 9\)

Step 4: Simplify the expression for part (b)

Calculate the value inside the parentheses first: \(7 \cdot 6 = 42\)Then subtract 9 from 42: \(42 - 9 = 33\)

Key Concepts

Substitution

Substitution is a fundamental technique in algebra where you replace a variable with a given numerical value. This is essential when you need to evaluate an expression at specific points. For example, in the exercise above, you are asked to evaluate the expression \(7x - 9\) for different values of \(x\).

• For part (a), \(x = 4\)


• For part (b), \(x = 6\)

The goal is to replace \(x\) with the given numbers and then simplify the expression to find the result. This process helps us understand how the value of the expression changes with different values of the variable.

Simplification

Simplification involves performing arithmetic operations to reduce an expression to its simplest form. After substitution, the next step is to simplify the expression. Let's look at how this is done:

First, substitute the value for \(x\) in the expression \(7x - 9\). For part (a), it becomes \(7(4) - 9\). Now, perform the multiplication: \(7 \times 4 = 28\).
Finally, subtract \(9\) from \(28\) to get \(19\). Always start by performing any operations inside parentheses, then follow the order of operations (PEMDAS/BODMAS).
By simplifying step by step, you ensure accuracy and a clear understanding of each part of the process.

Variable Substitution

Variable substitution is not just about plugging in numbers. It's about understanding what happens to an expression when the variable takes on different values. Consider the parts (a) and (b) of the exercise:

• For \(x = 4\), the expression \(7x - 9\) becomes \(7(4) - 9 = 19\).


• For \(x = 6\), the expression \(7x - 9\) becomes \(7(6) - 9 = 33\).

Seeing these changes help you understand how the output of an expression depends on the input values of the variable. Recognizing patterns in these changes is crucial for deeper skills in algebra.
Practicing variable substitution prepares you for more advanced topics, such as solving equations and understanding functions. So, always take your time to understand each step clearly.