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Chapter 2: Problem 9

Solve each equation, and check the solution. \(x-3=9\)

Short Answer

Expert verified
x = 12

Step by step solution

01

- Isolate the Variable

To isolate the variable x, add 3 to both sides of the equation. Equation: \(x - 3 = 9\)Add 3 to both sides:\(x - 3 + 3 = 9 + 3\)Simplifies to:\(x = 12\)
02

- Check the Solution

Substitute the solution back into the original equation to verify it is correct.Original Equation:\(x - 3 = 9\)Substitute \(x = 12\):\(12 - 3 = 9\)Since \(9 = 9\) is true, the solution is verified.

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Key Concepts

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

Isolating the Variable
To solve any linear equation, isolating the variable is the key first step. This means you want the variable by itself on one side of the equation.
Let's look at the example equation:
\(x - 3 = 9\).
Here, your goal is to isolate \(x\). To do this, you perform the same arithmetic operation on both sides of the equation to keep it balanced.
For this equation, you need to add 3 to both sides because \(x\) is being decreased by 3:
\(x - 3 + 3 = 9 + 3\)
This simplifies to:
\(x = 12\).
Remember, doing the same operation on both sides is crucial to maintaining the equality.
Checking Solutions
After finding the solution for \(x\), it's important to check if it's correct. This is called verifying your solution. To do this, you substitute your solution back into the original equation and see if it holds true.
Using our example where \(x = 12\), we substitute 12 back into the original equation
\(x - 3 = 9\).
Replace \(x\) with 12:
\(12 - 3 = 9\)
Then simplify: 9 = 9.
If both sides of the equation are equal, your solution is correct.
It's always a good practice to check your answers to avoid mistakes.
Step-by-Step Solutions
Breaking down a problem into step-by-step solutions makes it easier to tackle. Here's a breakdown of the steps:
  • Step 1: Identify what you need to isolate. In this example, it's the variable \(x\).
  • Step 2: Perform operations to get the variable alone. For \(x - 3 = 9\), add 3 to both sides to isolate \(x\).
  • Step 3: Simplify both sides to solve for the variable. Here, it simplifies down to \(x = 12\).
  • Step 4: Check your solution. Substitute \(12\) back into the original equation to see if the equation balances.
  • Result: If both sides match, your solution is correct.
Splitting a problem into these manageable steps can significantly help clarify and solve the equation efficiently.

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