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Chapter 2: Problem 25

Solve each equation, and check your solution. $$ x-8.4=-2.1 $$

Short Answer

Expert verified
The solution is x = 6.3.

Step by step solution

01

- Isolate the Variable

Add 8.4 to both sides of the equation to get the variable by itself on one side. \[ x - 8.4 + 8.4 = -2.1 + 8.4 \] this simplifies to \[ x = 6.3 \]
02

- Check the Solution

Substitute the value of x back into the original equation to ensure it satisfies the equation.Original equation: \[ x - 8.4 = -2.1 \]Substitute x = 6.3: \[ 6.3 - 8.4 = -2.1 \]Simplify the left side: \[ -2.1 = -2.1 \]Since both sides of the equation are equal, the solution x = 6.3 is correct.

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Key Concepts

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

Isolate the Variable
To solve any equation, your main goal is to isolate the variable. This means getting the variable by itself on one side of the equation.
Let's take a look at our exercise: \( x - 8.4 = -2.1 \). To isolate x, we need to undo what is being done to it. Here, x is being subtracted by 8.4. To reverse this, we do the opposite operation, which is adding 8.4 on both sides of the equation:
  • \( x - 8.4 + 8.4 = -2.1 + 8.4 \)
      • By performing this step, we cancel out the -8.4 on the left, ending up with: \( x = 6.3 \)Jessica To quickly isolate the variable, always consider what operation is performed on it and do the opposite to both sides of the equation.
Check the Solution
Once you find the value of the variable, it's essential to check if your solution is correct. This confirmation step ensures you didn't make a mistake during your calculations. Take the value you found and substitute it back into the original equation.
Our original equation is: \( x - 8.4 = -2.1 \). We found that: \( x = 6.3 \)
Now we'll substitute 6.3 back into the equation:
  • 6.3 - 8.4
    • equals
  • -2.1
When you simplify, both sides of the equation should match:
  • \( -2.1 = -2.1 \).
Since they are equal, we can confidently say our solution, \( x = 6.3 \), is correct.
Simplify Both Sides
Simplifying equations is crucial in solving linear equations. It makes the equation easier to work with and helps eliminate any potential errors.
In the given exercise, initially, we didn't have a complex expression, but let's break down why simplifying each side is essential.
Starting with our equation:
  • \( x - 8.4 = -2.1 \)
    • We aim to isolate x by adding 8.4 on both sides.
    • \( x - 8.4 + 8.4 = -2.1 + 8.4 \).

        • Simplifying here means combining like terms:
        On the left side, \( -8.4 + 8.4 \) simplifies to 0, leaving us with x.
        • On the right side, \( -2.1 + 8.4 \) simplifies to 6.3.

            So, ensuring both sides of the equation are as simple as possible helps us correctly isolate and solve for the variable.

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