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Chapter 2: Problem 3

Evaluate each expression if \(x=6\). a. \(2 x+3\) b. \(2(x+3)\) c. \(5 x-13\) d. \(\frac{x+9}{3}\)

Short Answer

Expert verified
a. 15, b. 18, c. 17, d. 5.

Step by step solution

01

Substitute in Part (a)

Given the expression \(2x + 3\). Substitute \(x = 6\) into the expression: \(2(6) + 3\).
02

Calculate Part (a)

Evaluate the expression: \(2(6) + 3 = 12 + 3 = 15\).
03

Substitute in Part (b)

Given the expression \(2(x+3)\). Substitute \(x = 6\) into the expression: \(2(6 + 3)\).
04

Calculate Part (b)

Evaluate the expression: \(2(6 + 3) = 2(9) = 18\).
05

Substitute in Part (c)

Given the expression \(5x - 13\). Substitute \(x = 6\) into the expression: \(5(6) - 13\).
06

Calculate Part (c)

Evaluate the expression: \(5(6) - 13 = 30 - 13 = 17\).
07

Substitute in Part (d)

Given the expression \(\frac{x+9}{3}\). Substitute \(x = 6\) into the expression: \(\frac{6+9}{3}\).
08

Calculate Part (d)

Evaluate the expression: \(\frac{6+9}{3} = \frac{15}{3} = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a fundamental technique used in evaluating algebraic expressions. This method involves replacing the variable in an expression with a specific number, which is given in the problem.

In the original exercise, we are asked to evaluate various expressions by substituting the value of \( x = 6 \) into each expression. Let's break down how substitution works with a specific example:
  • Take the expression \( 2x + 3 \). When using substitution, replace \( x \) with \( 6 \). This gives \( 2(6) + 3 \).
  • The outcome is a new expression with numbers only, making it easier to compute the final result.
The substitution method simplifies the evaluation process, making it straightforward to solve expressions by directly plugging in the given numbers. It helps to convert algebraic to arithmetic problems.
Simplifying Expressions
Simplifying expressions is another crucial step when evaluating. Simplification makes complicated algebraic expressions into a form that's easier to understand and compute.

Let's explore how simplification occurs after substitution, using one of the exercise examples:
  • Consider the expression \( 2(6) + 3 \) after substituting \( x = 6 \). We simplify this by multiplying \( 2 \) and \( 6 \) to get \( 12 \), and then adding \( 3 \) to result in \( 15 \).
  • In each step, perform operations as per the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), commonly known as PEMDAS.
By consistently applying these operations, expressions get reduced to their simplest numeric form, allowing for easy calculation and solution presentation.
Algebraic Evaluation
Algebraic evaluation refers to the process of finding the value of an algebraic expression. This involves using substitution and simplification to solve expressions for a given value of the variable.

The primary goal of algebraic evaluation is to achieve a numeric outcome from an expression. For instance, in part \( c \) of the exercise, starting with the expression \( 5x - 13 \):
  • You first substitute \( x = 6 \) into the expression, turning it into \( 5(6) - 13 \).
  • Next, simplify this to \( 30 - 13 \), and then, perform the subtraction to arrive at the final result of \( 17 \).
By following these logical steps, the original algebraic problem gets translated into a straightforward arithmetic problem. This ensures a thorough understanding of the underlying mathematical principles.

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Most popular questions from this chapter

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