Problem 68 - 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 \(x .\) $$7 x-2+4 x-1$$

Step-by-Step Solution

Answer

The expression evaluates to 30 when \(x = 3.\)

Step 1: Rewrite the Expression

First, look at the expression given in the problem: \[7x - 2 + 4x - 1\] Our job is to find the value when \(x = 3\). Decide if you want to substitute \(x\) first or simplify first. In this solution, we will simplify the expression before substituting.

Step 2: Combine Like Terms

In the expression \(7x - 2 + 4x - 1\), we can combine like terms. Like terms are terms that have the same variable raised to the same power. Combine the \(x\) terms: \((7x + 4x) = 11x\). Next, combine the constant numbers: \((-2 - 1) = -3\). Thus, the simplified expression is: \[11x - 3\]

Step 3: Substitute the Value of x into the Simplified Expression

Now substitute \(x = 3\) into the simplified expression \(11x - 3\). Replace \(x\) with \(3\): \[11(3) - 3\].

Step 4: Perform the Arithmetic Calculation

Complete the calculation by performing the multiplication and subtraction: First, calculate \(11(3)\) to get \(33\). Then subtract 3 from 33: \[33 - 3 = 30\]. The calculation results in 30.

Key Concepts

Combining Like Terms

When dealing with algebraic expressions, combining like terms is an essential skill. It means grouping terms that look alike, specifically those with the same variable raised to the same power.
In the exercise, we have the expression:

• \(7x - 2 + 4x - 1\)

The terms \(7x\) and \(4x\) are like terms because they both include the variable \(x\). Similarly, the numbers \(-2\) and \(-1\) are like terms as they are constants.
Here's how you combine them:

• Add the coefficients of the like terms: \(7x + 4x = 11x\)
• Combine the constants: \(-2 - 1 = -3\)

After combining like terms, the expression is simplified to:

• \(11x - 3\)

Algebraic Substitution

Algebraic substitution is the process of replacing a variable with a number. This step is how we calculate the expression's value for a specific variable value.
In the given exercise, the problem requires evaluating the expression when \(x = 3\).
After simplifying the expression, we get:

• \(11x - 3\)

Substitute \(x = 3\) into this expression. This means replacing each occurrence of \(x\) with the number 3.

• The expression becomes \(11(3) - 3\)

Now, you're ready to solve the expression with the substituted values. This step is crucial because it translates the general expression into a numerical form, allowing you to perform arithmetic calculations.

Simplifying Expressions

Simplifying expressions involves reducing them into the simplest or most compact form. It's a critical task for making arithmetic with algebraic expressions more straightforward.
This process often includes:

• Combining like terms
• Performing operations in an orderly manner, often following PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)

In the example:

• We first combined like terms to obtain \(11x - 3\)
• Then substituted \(x = 3\) to get \(11(3) - 3\)

Finally, perform the arithmetic:

• Multiply and subtract: \(11 \times 3 = 33\), then \(33 - 3 = 30\)

Simplifying expressions helps us to see the structure of problems more clearly, and makes calculations much easier by minimizing complexity.