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Chapter 2: Problem 43

Solve each equation, and check your solution. $$ 1.4 x-3=0.4 x $$

Short Answer

Expert verified
x = 3

Step by step solution

01

Move variable terms to one side

Subtract 0.4x from both sides of the equation to get the variable terms on one side: 1.4x - 0.4x - 3 = 0.4x - 0.4xThis simplifies to: 1.0x - 3 = 0
02

Isolate the variable

To isolate the variable, add 3 to both sides of the equation: 1.0x - 3 + 3 = 0 + 3This simplifies to: x = 3
03

Check the solution

Substitute the value of x back into the original equation to verify it: 1.4(3) - 3 = 0.4(3)Calculate each side: 4.2 - 3 = 1.2 and 0.4(3) = 1.2Both sides are equal, which verifies that x = 3 is the correct solution.

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Key Concepts

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

Understanding the Process of Solving Equations
Solving linear equations involves finding the value of the variable that makes the equation true. To start, always ensure the equation is properly set up. Linear equations have the format \(a x + b = c\), where \(a\), \(b\), and \(c\) are constants. Each step in solving these equations aims to simplify the equation until the variable is isolated. Here, the original equation is:
1.4x - 3 = 0.4x.
The goal is to find the value for \(x\) that makes this equation true. Throughout the process, we will perform operations that keep the equation balanced.
Isolating the Variable in the Equation
Variable isolation is a key step in solving equations. It involves rearranging the equation to get the variable on one side and the constants on the other.
Start with: 1.4x - 3 = 0.4x.
To isolate the variable \(x\), follow these steps:
  • Subtract 0.4x from both sides: 1.4x - 0.4x - 3 = 0
  • Simplify: 1x - 3 = 0
  • Add 3 to both sides to move the constant: 1.0x - 3 + 3 = 0 + 3
  • After simplifying further, we get: x = 3
Now, \(x\) is isolated and equals to 3.
Verifying the Solution
Verification ensures that our solution is correct. Substitute the value obtained back into the original equation.
Here's the original equation: 1.4x - 3 = 0.4x.
Replace \(x\) with 3:
  • Left side: 1.4(3) - 3 = 4.2 - 3 = 1.2
  • Right side: 0.4(3) = 1.2
Both sides equal 1.2, confirming our solution is correct. Always double-check your solutions to avoid mistakes and ensure full understanding. By verifying, you confirm that the value you found for the variable truly satisfies the original equation.

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