Problem 99 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$\frac{1}{3}(x-2)=\frac{1}{5}(x+4)+3 ; 26$$

Step-by-Step Solution

Answer

No, 26 is not a solution of the equation, as the left side of the equation (8) is not equal to the right side (9).

Step 1: Substituting Value

Substitute the value \( x = 26 \) into the given equation.

Step 2: Simplify Left Side

First, let's simplify the left side of the equation. The left side of the equation, \( \frac{1}{3}(x-2) \), becomes \( \frac{1}{3}(26-2) \) which simplifies to 8.

Step 3: Simplify Right Side

Now, let's simplify the right side of the equation. The right side of the equation, \( \frac{1}{5}(x+4)+3 \), becomes \( \frac{1}{5}(26+4)+3 \), which simplifies to 9.

Step 4: Comparing Left and Right Sides

We then compare the newly obtained right and left sides of the equation. If they are equal, then 26 is a solution to the equation, otherwise it is not.

Key Concepts

Substitution Method

In algebra, the substitution method is a powerful technique for solving equations. It involves replacing variables with given values to see if the equation holds true. This method helps to verify if a specific number is indeed a solution to the equation. For example, in the exercise provided, we are given the equation:
\[\frac{1}{3}(x-2) = \frac{1}{5}(x+4) + 3,\]and asked to determine if 26 is a solution. The first step is to replace \( x \) with 26 in both sides of the equation. By doing so, we evaluate the expression to check its validity. It’s like taking a proposed answer and testing it to see if it "fits" into the equation. This method is straightforward, allowing us to quickly plug in values and simplify each side of the equation separately. The substitution method provides a systematic way to check potential solutions efficiently and effectively.

Equation Simplification

Equation simplification is key to solving algebraic problems accurately. It involves reducing complex equations to simpler forms, making them easier to interpret and solve. After substitution, as in our exercise, we need to simplify each side of the equation:
- For the left side: \( \frac{1}{3}(x-2) \) with \( x = 26 \) becomes \( \frac{1}{3}(26-2) \). Simplifying this gives us 8.- For the right side: \( \frac{1}{5}(x+4) + 3 \) with \( x = 26 \) becomes \( \frac{1}{5}(26+4) + 3 \). Simplifying this gives us 9.Simplifying ensures that both sides of the equation are in their simplest numerical form, allowing us to accurately compare them. This process often involves arithmetic operations like addition, subtraction, multiplication, and division. Simplification not only aids in solving equations but also enhances our understanding of the problem at hand.

Solution Verification

Solution verification is the final step in the problem-solving process. After substituting and simplifying, we need to compare the simplified results from each side of the equation to determine if the proposed solution is correct.
For our exercise, after simplifying, the left side equals 8 and the right side equals 9. Since these values are not equal, the number 26 is not a solution to the equation. Verification is crucial in confirming whether our results align with the original equation.
It acts as a final check to ensure accuracy in our calculations and reasoning. This step prevents errors by allowing us to re-evaluate our work and ensure every step makes logical sense. Solution verification is an essential part of developing robust problem-solving skills, reinforcing the accuracy and precision required in algebra.