Problem 96 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$12-\frac{1}{4} z=\frac{1}{2} z ; 20$$

Step-by-Step Solution

Answer

No, the number 20 is not a solution to the equation \(12 - \frac{1}{4}z = \frac{1}{2}z\).

Step 1: Substitute the given number for \(z\)

Insert the number 20 wherever \(z\) appears in the equation: \(12 - \frac{1}{4}*20 = \frac{1}{2}*20\)

Step 2: Simplification

Simplify both sides of the equation by carrying out the operation. On the left, we have \(12 - 5 = 7\) while on the right, it simplifies to \(10\)

Step 3: Comparing

Compare both sides of the equation. Since they are not equal (\(7 \neq 10\)), we conclude that 20 is not a solution to the equation.

Key Concepts

Substitution Method

The substitution method is an essential technique when solving equations. It involves replacing variables with specific values to see if the equation holds true. In our example, we want to determine if the number 20 is a solution to the equation \(12 - \frac{1}{4} z = \frac{1}{2} z\). By substituting 20 for \(z\), the equation transforms into a purely numerical form: \(12 - \frac{1}{4} \cdot 20 = \frac{1}{2} \cdot 20\).

This method allows us to focus on the simplification of numerical expressions, making it easier to compare both sides of the equation. Remember to correctly replace all instances of the variable to avoid errors. Substitution is a powerful tool in algebra that often serves as a first step in solving for a specific variable or verifying possible solutions.

Simplification of Equations

Simplification is the process of breaking down equations into simpler forms. Once we have substituted the given number into the equation, the next step is to carry out basic operations to simplify each side.

In our example, after substituting 20 for \(z\), we simplify the left side: \(12 - \frac{1}{4} \cdot 20\) becomes \(12 - 5\), which results in 7. On the right side, \(\frac{1}{2} \cdot 20\) simplifies directly to 10.

By simplifying, we reduce complex equations to manageable expressions, making it easier to interpret and compare them. Follow these steps closely:

• Perform any multiplications and divisions first.
• Next, handle additions and subtractions.
• Keep calculations step-by-step to avoid mistakes.

This methodical approach is crucial for keeping equations simple and understandable.

Comparing Solutions

Comparing solutions is a crucial part in verifying if the substituted value turns the equation into a true statement. After simplification, we end up with two separate numbers on either side of the equation. In our case, we have 7 on the left and 10 on the right.

The key question is: Do these numbers equal each other? By checking equality, we confirm whether or not the initial number is a valid solution. For this particular equation, since 7 is not equal to 10, we confidently determine that 20 is not a solution.

This comparative step helps us validate solutions efficiently. Important things to note:

• Ensure both sides are fully simplified before comparing.
• If both sides match, the value is a solution.
• If they do not match, the value isn't a solution.

Mastering this skill allows for effective troubleshooting and problem-solving in algebra.