Problem 18 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$3 r^{2}-17 \text { when } r=6$$

Step-by-Step Solution

Answer

The result of the expression \(3r^2 - 17\) where \(r=6\) is \(91\).

Step 1: Substitute the Value of r

Replace \(r\) in the expression \(3r^2 - 17\) with the given value \(r=6\). This results in the expression \(3(6)^2 - 17\).

Step 2: Apply the Order of Operations - Exponent

First, handle the exponent in the expression, resulting in \(3(36) - 17\).

Step 3: Apply the Order of Operations - Multiplication

Next, handle the multiplication in the expression, resulting in \(108 - 17\).

Step 4: Apply the Order of Operations - Subtraction

Finally, perform the subtraction to simplify the expression to a single number, \(91\).

Key Concepts

Order of Operations

Evaluating algebraic expressions correctly requires an understanding of the Order of Operations. This is a fundamental concept that dictates the sequence in which different operations, such as addition, subtraction, multiplication, and exponents, should be performed. The common mnemonic to remember this order is PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order is crucial in ensuring that everyone arrives at the same result when evaluating an expression. For example, in the expression \(3(6)^2 - 17\), you must first evaluate the exponent \((6^2)\), then the multiplication \((3 \times 36)\), and finally, the subtraction \(108 - 17\). Understanding and applying the Order of Operations correctly can prevent many errors in math calculations.

Substitution

Substitution in algebra is the process of replacing a variable in an expression with a given value. This technique is helpful in evaluating expressions, especially when a specific numerical answer is needed. In our presented expression \(3r^2 - 17\), the variable \(r\) needs to be substituted with the value 6. So, every instance of \(r\) is replaced, resulting in a modified expression \(3(6)^2 - 17\). Substituting allows us to transform abstract math expressions into concrete calculations.A key tip when substituting is to write parentheses around the substituted numbers, particularly when dealing with exponents or multiple operations. This prevents mistakes and keeps calculations clear. The step of substitution lays the groundwork for further evaluation using the Order of Operations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, and division). These can be as simple as a single term, or more complex with multiple terms.Consider the expression \(3r^2 - 17\), which includes both a variable term \(3r^2\) and a constant term \(-17\). When dealing with algebraic expressions:

• Understand the role of each component: variables stand for unknown or changeable values, coefficients give weight to these variables, and constants are fixed values.
• Recognize different types:
  • Monomials (single term, e.g., \(3r^2\))
  • Binomials (two terms, e.g., \(3r^2 - 17\))
  • Polynomials (three or more terms)

By mastering the parts of an algebraic expression and how they interact, you can confidently evaluate and manipulate expressions. This basis helps in higher mathematical concepts and problem-solving.