Problem 62 - 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 \(x .\) $$6 x+8 x+7$$

Step-by-Step Solution

Answer

The value of the expression is 49 when \(x = 3\).

Step 1: Understand the Expression

We are given the algebraic expression \(6x + 8x + 7\). Our task is to substitute \(x=3\) into this expression and calculate the result.

Step 2: Simplify the Expression

First, we combine like terms in the expression to simplify it. The terms \(6x\) and \(8x\) are like terms. So, we add them together: \[6x + 8x = (6 + 8)x = 14x\]Thus, the expression becomes \(14x + 7\).

Step 3: Substitute the Value of x

Next, we substitute \(x = 3\) into the simplified expression \(14x + 7\): \[14(3) + 7\]

Step 4: Perform the Multiplication

Calculate the multiplication in the expression:\[14 imes 3 = 42\]

Step 5: Add the Result

Add the result from the multiplication to 7:\[42 + 7 = 49\]

Step 6: Conclude the Calculation

The final calculated value of the expression \(14x + 7\) when \(x = 3\) is 49.

Key Concepts

Combining Like Terms

To simplify an algebraic expression, one primary approach is to combine like terms. Like terms are terms in an expression that have exactly the same variable(s) and exponent(s). For example, in the expression \(6x + 8x + 7\), the terms \(6x\) and \(8x\) are like terms. They both contain the variable \(x\) raised to the same power of 1.
Combining like terms means adding or subtracting their coefficients while retaining the common variable part. In our example, we combine \(6x\) and \(8x\) to get \((6 + 8)x = 14x\). This simplification makes the expression easier to work with later steps.

• Identify like terms by looking for the same variable part.
• Add or subtract their coefficients.
• Simplify the expression by combining the terms.

After combining, our expression becomes \(14x + 7\), which is a more straightforward form.

Substitution

Substitution involves replacing a variable in an expression with a specific value to find the value of the expression. In our expression, once we have simplified to \(14x + 7\), we substitute \(x = 3\) to calculate the numeric value.
The substitution step is straightforward: simply replace \(x\) with \(3\) in each instance it appears in the expression. So, \(14x + 7\) becomes \(14(3) + 7\).

• Identify where the variable occurs in the expression.
• Replace each occurrence with the given number.
• Ensure all substitutions are consistent throughout the expression.

This step converts the algebraic expression into an arithmetic one that can be easily evaluated.

Simplifying Expressions

Simplifying expressions is about reducing them to their most manageable form before performing calculations. This involves combining like terms, performing arithmetic operations, and ensuring the expression is as straightforward as possible.
In our example, this means simplifying \(6x + 8x + 7\) to \(14x + 7\) by combining like terms. With substitution, we then have \(14(3) + 7\). Now, it's time to do the arithmetic:

• First, multiply: \(14 \times 3 = 42\).
• Second, add the constant term: \(42 + 7 = 49\).

This calculation gives us the final value of 49. Simplifying an expression beforehand makes these operations straightforward and helps avoid mistakes. Always aim to carry out simplification to gain clarity and ease when solving problems.