Problem 13 - Beginning and Intermediate Algebra

Question

Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(5 x-4\)

Step-by-Step Solution

Answer

For \(x=4\): 16, and for \(x=6\): 26.

Step 1: Substitute for (a)

First, substitute the value of \(x = 4\) into the expression \(5x-4\). This gives us \[5(4) - 4\].

Step 2: Simplify for (a)

Next, simplify the expression by multiplying and subtracting: \[5(4) - 4 = 20 - 4 = 16\].

Step 3: Substitute for (b)

Now, substitute the value of \(x = 6\) into the expression \(5x-4\). This gives us \[5(6) - 4\].

Step 4: Simplify for (b)

Finally, simplify the expression by multiplying and subtracting: \[5(6) - 4 = 30 - 4 = 26\].

Key Concepts

Substitution

Substitution in algebra involves replacing variables with their given numerical values. When you substitute, you plug the values provided directly into the expression. For example, if you're given the expression \(5x - 4\) and you need to evaluate it for \(x = 4\), you'll replace every instance of \(x\) with 4. This turns \(5x - 4\) into \(5(4) - 4\). Substitution is a key skill in algebra because it helps us simplify and evaluate expressions.

Simplification

Simplification is the process of performing the arithmetic operations to reduce an expression to its simplest form. Using the same example \(5(4) - 4\), after substituting the value of \(x\), you perform the operations:

• First, multiply 5 by 4 to get 20.
  
• Then, subtract 4 from 20 to get 16.
  

This process has simplified the expression \(5(4) - 4\) to 16. Simplification helps by making complex expressions easier to understand and solve.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. An example is \(5x - 4\). These expressions can take different forms and can be evaluated by substitution followed by simplification. Understanding how to manipulate algebraic expressions is essential in solving algebra problems. Practicing with different values of variables helps in mastering the art of working with algebraic expressions. By recognizing patterns and operations in these expressions, students can solve a wide array of algebra problems more effectively.