Problem 8 - Beginning and Intermediate Algebra

Question

The value of \(\frac{x+3}{5}\) for \(x=10\) was found incorrectly as follows. $$ \begin{array}{l} \frac{x+3}{5} \\ =\frac{10+3}{5} \\ =2+3 \\ =5 \end{array} $$ Find the correct value of the expression.

Step-by-Step Solution

Answer

The correct value is 2.6.

Step 1 - Substitute x in the Expression

Substitute \(x = 10\) into the expression \( \frac{x+3}{5} \). This gives us: \( \frac{10+3}{5} \).

Step 2 - Simplify the Numerator

Add the numbers in the numerator. \(10 + 3 = 13\). So the expression becomes \( \frac{13}{5} \).

Step 3 - Perform the Division

Divide the numerator by the denominator. \( \frac{13}{5} = 2.6 \).

Key Concepts

Evaluating Algebraic Expressions

Evaluating algebraic expressions is fundamental in algebra. It involves replacing variables with numbers and performing arithmetic operations to find the value of the expression.
Let's break this down using the expression given in the exercise: \( \frac{x+3}{5} \). Here, x is the variable, and we need to find its value when x equals 10.
This substitution translates to: \( \frac{10+3}{5} \). Each part of the expression must be simplified methodically. Substitution is straightforward but requires careful attention to arithmetic operations to avoid mistakes.

Substitution Method

The substitution method in algebra involves replacing variables with their given values.
Here's a step-by-step breakdown for this exercise:
1. **Identify the variable.** In the expression \( \frac{x+3}{5} \), the variable is x.
2. **Substitute the value of the variable.** We are given x=10. So substitute 10 for x, leading us to \( \frac{10+3}{5} \).
This simplifies the first part of the problem.
The key is to ensure accuracy during substitution. Mistakes here can lead to entirely incorrect results further down the steps.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. They are the building blocks of higher-level math like algebra.
In our example, we perform the following arithmetic steps:
1. **Addition in the numerator:** \( 10 + 3 = 13 \). This step transitions our expression to \( \frac{13}{5} \).
2. **Division:** Now divide the numerator by the denominator. \( \frac{13}{5} = 2.6 \).
This gives us the final correct value of the expression. Remember, each operation must be done in the correct sequence to avoid calculation errors. Algebra, while seemingly complex, relies heavily on these basic operations.