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Chapter 2: Problem 46

Solve equation, and check your solution. \(\frac{x}{5}=15\)

Short Answer

Expert verified
The solution is \(x = 75\).

Step by step solution

01

Understand the Equation

The equation given is \(\frac{x}{5}=15\). This means that some number divided by 5 equals 15. Our goal is to find that number.
02

Isolate the Variable

To solve for \(x\), we need to remove the fraction. We can do this by multiplying both sides of the equation by 5:
03

Performing the Multiplication

Multiply both sides by 5 to get \[ \frac{x}{5} \times 5 = 15 \times 5 \] which simplifies to \[ x = 75 \].
04

Check the Solution

To ensure our solution is correct, we substitute \(x = 75\) back into the original equation: \[ \frac{75}{5} = 15 \]. This simplifies to 15 = 15, which is true. Hence, the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable
When solving linear equations, the first step is often to isolate the variable. Here, we have the equation \(\frac{x}{5}=15\). The goal is to get the variable \ x \ by itself on one side of the equation. Start by identifying what operation is being performed on the variable. In this equation, \ x \ is being divided by 5. To isolate \ x \, you need to perform the opposite operation, which is multiplication. By multiplying both sides of the equation by 5, you effectively cancel out the division on the left side.
Multiplication Property of Equality
The multiplication property of equality is crucial when solving equations. It states that if you multiply both sides of an equation by the same number, the two sides remain equal. Applying this to our equation \(\frac{x}{5}=15\), we multiply both sides by 5: \[ \frac{x}{5} \times 5 = 15 \times 5 \] This simplifies to \[ x = 75 \] Multiplying by the same number on both sides helps to maintain the balance of the equation, which is essential for finding the correct value of the variable.
Verification of Solutions
After isolating the variable and solving for it, the final step is to verify the solution. Verification ensures that the solution is correct and satisfies the original equation. For our example, substitute \ x = 75 \ back into the initial equation: \[ \frac{75}{5} = 15 \] This simplifies to \[ 15 = 15 \] Since both sides of the equation are equal, our solution is verified. Verification is an important step as it confirms that no errors were made during the solving process and that the solution is indeed accurate.

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