Problem 63 - Prealgebra

Question

Find the value of each of the following expressions when \(x=3 .\) You may substitute 3 for \(x\) in each expression the way it is written, or you may simplify each expression first and then substitute 3 for \(x .\) $$9 x+x+3+7$$

Step-by-Step Solution

Answer

The value of the expression is 40.

Step 1: Understand the expression

The expression we need to solve is \(9x + x + 3 + 7\). We are asked to find its value when \(x = 3\).

Step 2: Simplify the expression

First, combine like terms in the expression. Both \(9x\) and \(x\) are like terms. This simplifies the expression to \(10x + 3 + 7\).

Step 3: Simplify the constant terms

Now, add the constant terms \(3\) and \(7\), which results in \(10\). The expression now becomes \(10x + 10\).

Step 4: Substitute the value of \(x\)

Substitute \(x = 3\) into the simplified expression \(10x + 10\). This gives us \(10(3) + 10\).

Step 5: Calculate the result

Multiply \(10\) by \(3\) to get \(30\), and then add \(10\) to get \(40\). Therefore, the value of the expression when \(x = 3\) is \(40\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It helps us transform complex expressions into simpler forms, making them easier to work with. The goal is to rewrite the expression in its simplest form. In our exercise, we started with the expression \(9x + x + 3 + 7\). This expression is cluttered with multiple terms, including variables and constants. By simplifying, we aim to combine all like terms and get rid of any redundancies.

• Identify like terms, which in our case are the terms involving \(x\): \(9x\) and \(x\).
• Combine these terms to reduce the expression to a cleaner form: \(10x\).
• Look at the constants, \(3\) and \(7\), and sum them to further simplify: \(10\).

This simplification process results in a much neater expression: \(10x + 10\). By reducing the number of terms and operations, we make it easier to perform further algebraic manipulations, such as evaluating its value for a given variable.

Combining Like Terms

Combining like terms is a fundamental technique used to condense algebraic expressions. Like terms are terms that have the same variable raised to the same power. In the exercise, we encountered the terms \(9x\) and \(x\). Both of these terms are like terms because they involve the same variable \(x\) raised to the same power (which is 1, since no exponent is shown).

• Add the coefficients of the like terms directly. For example, \(9x + x\) becomes \(10x\), because \(1x\) is the same as \(x\).
• The combination process streamlines the expression, enabling easier calculations.

This technique is not only used to simplify, but also critical in solving equations. By mastering combining like terms, you develop a key proficiency that applies across different areas of algebra. This skill is particularly useful when manipulating expressions that are to be solved or evaluated.

Evaluating Algebraic Expressions

Evaluating algebraic expressions involves substituting a specific value for the variable and calculating the result. It's like solving a puzzle where each variable is replaced by a given number. In our exercise, we needed to find the expression's value at \(x = 3\).

• Start by substituting the given value into the simplified expression. Here, \(10x + 10\) becomes \(10(3) + 10\).
• Proceed to perform the arithmetic operations: Multiply \(10\) by \(3\) to get \(30\), then add \(10\) resulting in \(40\).

Evaluating expressions is a straight-forward process once you've simplified them. This method is beneficial for verifying solutions and understanding how expressions change based on variable values. It is a practical application that appears in various math problems, emphasizing the necessity of accuracy in arithmetic and algebraic manipulation.