Problem 54 - Algebra A Combined Function

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5 .\) $$ 6 y-8 $$

Step-by-Step Solution

Answer

The expression evaluates to 10.

Step 1: Substitute the Values

First, substitute the given value of \( y = 3 \) into the expression \( 6y - 8 \). This gives us \( 6(3) - 8 \).

Step 2: Multiply

Next, multiply the 6 by 3 to get 18. Our expression now looks like \( 18 - 8 \).

Step 3: Subtract

Finally, subtract 8 from 18 to get the result: \( 10 \).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations such as addition, subtraction, multiplication, and division. For example, in the expression \(6y - 8\), "6y" represents a term where 6 is multiplied by a variable \(y\). The "-8" is also a term, which is a constant number. Together, they form the algebraic expression.

These expressions allow us to represent general relationships in mathematics and can be simplified or evaluated by substituting values in place of the variables. Variables in expressions are like placeholders that can take different values in different situations. By substituting numbers for these variables, we can solve or evaluate the expression to find a specific value.

Evaluation of Expressions

Evaluation of expressions involves replacing variables with given numerical values and then simplifying the expression using arithmetic operations. This process helps in finding out what an expression equals when specific values are assigned to variables.

Let's see the process with an example. Given the expression \(6y - 8\) and knowing that \(y = 3\), we substitute 3 for \(y\) to evaluate the expression. It becomes \(6(3) - 8\). This substitution is the first step in evaluating.

After substitution, the expression is simplified by performing the arithmetic operations as indicated. It's important to carry out the operations step by step in the correct sequence to ensure accurate results. This leads us to another crucial concept known as the "Order of Operations".

Order of Operations

The order of operations is a rule used to clarify which procedures to perform first in a given mathematical expression. It's crucial for solving expressions correctly, especially when they include multiple operations. The common acronym for remembering the order is PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• M/D: Multiplication and Division (left to right)
• A/S: Addition and Subtraction (left to right)

In the exercise example, once we substitute \(y\) with 3 in \(6y - 8\), we then apply the order of operations by first handling the multiplication part, multiplying 6 by 3. This results in 18. We then proceed to subtraction, subtracting 8 from 18 to achieve the final value, which is 10.

By following the order of operations, we ensure expressions are simplified and evaluated correctly and efficiently.