Problem 78 - Algebra 1

Question

Check whether the given number is a solution of the equation. $$3 w-7=w+1 ; 5$$

Step-by-Step Solution

Answer

No, 5 is not a solution for the equation \(3w-7=w+1\).

Step 1: Write down the given equation

The given equation is \( 3w-7=w+1 \). The aim is to check whether the number 5 is a solution of this equation.

Step 2: Substitute 5 for \(w\)

In this step, \(w\) is substituted by 5 in the equation: \( 3(5) - 7 = 5 + 1 \). Thus, the equation becomes \( 15 - 7 = 5 + 1 \).

Step 3: Simplify both sides of the equation

If we simplify both sides of the equation, we get \( 8 = 6 \). The left side doesn't equal the right side.

Step 4: Summarize the Results

Since the left side doesn't equal the right side after substituting \(w\) = 5 in the equation, it means that 5 is not a solution for the equation \( 3w-7=w+1 \).

Key Concepts

Substitution Method

The substitution method is a way to solve equations by replacing variables with numbers. It simplifies the process of verifying if a particular value is a solution to the equation.
In the context of our exercise, we substitute the variable \( w \) with the number 5. Essentially, you're testing to see if setting \( w = 5 \) makes the equation hold true. Here's how you do it:
1. Identify the variable in the equation. For example, \( 3w-7=w+1 \) has \( w \) as the variable.2. Substitute the given number for the variable. Replace \( w \) with 5, resulting in \( 3(5) - 7 = 5 + 1 \).3. Perform the arithmetic operation to check if both sides of the equation are equal.

• Use substitution to directly test potential solutions.
• Helps simplify and validate your equation.

This method is straightforward because it allows you to directly verify if the substitution yields equal expressions and therefore, solves the equation.

Checking Solutions

Checking solutions is an important part of solving equations. It confirms whether the substituted value makes the equation true.
After making the substitution, as seen in our example, simplifying both sides of the equation is crucial to check the validity of the substituted value.
To check the solution:

• Simplify both sides of the equation after substitution.
• Compare the two sides to see if they are equal.
• If they are equal, the number is a solution.
• If not, the number is not a solution.

In the example, substituting 5 into the equation and simplifying leads us to \( 8 eq 6 \). Since the two sides are not equal, 5 is not a solution. This process ensures accuracy in solving equations.

Equation Simplification

Equation simplification involves reducing an equation to its simplest form. This typically involves performing basic arithmetic operations to make the equation easier to work with.
Simplifying an equation is essential for effectively comparing both sides of the equation during solution checking.
Steps involved in simplification:

• Handle arithmetic operations like addition, subtraction, multiplication, or division.
• Carefully transition through each step like \( 15 - 7 = 5 + 1 \).
• Bring both sides down to single numbers or expressions for easier comparison.

Our exercise involves simplifying both \( 15 - 7 \) and \( 5 + 1 \), leading to \( 8 \) and \( 6 \), respectively. Despite simplifying, when the resulting numbers don't match, it indicates that the original substituted value isn't a solution.