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When solving a linear equation, the primary goal is to find the value of the variable that makes the equation true. To do this, we need to isolate the variable on one side of the equation. Think of it as gathering all terms with the variable together and shedding extra numbers that don't contain the variable. In practice, this often involves performing the opposite operation of what is currently on the variable's side. For example, if there is addition as in the equation \( 7 + 4x = 29 \), you should subtract 7 from both sides. This action maintains the balance of the equation while simplifying one side:

• Starting equation: \( 7 + 4x = 29 \)
• Subtract 7 from both sides: \( 7 + 4x - 7 = 29 - 7 \)
• Simplified result: \( 4x = 22 \)

By isolating the variable, you've successfully reduced the equation into a simpler form that's easier to solve. Remember, maintaining balance by doing the same operation on both sides is key!

Once the variable has been isolated, the next step is to further simplify the equation. Simplification is all about making the equation easier to work with by reducing it to its simplest possible form. For our example, after isolating, we have the equation \( 4x = 22 \). At this point, it's a simple matter of dealing with the multiplication of 4 on the variable:

• Equation: \( 4x = 22 \)
• To simplify, divide both sides by 4: \( \frac{4x}{4} = \frac{22}{4} \)
• This yields: \( x = \frac{22}{4} \)

Equation simplification helps streamline your work and removes complexity, helping you find the value of the variable effectively. It's about breaking down operations and dealing with them step-by-step until you get to a cleaner result.

Fraction simplification is an important step in solving equations when fractions are involved. This process is about reducing fractions to their simplest form, which involves dividing the numerator and the denominator by their greatest common divisor (GCD). In our case, after dividing through in the steps of solving the equation \( 4x = 22 \), we have the fraction \( \frac{22}{4} \). To simplify:

• Identify the GCD of 22 and 4, which is 2.
• Divide both the numerator and denominator by the GCD (2): \( \frac{22 \div 2}{4 \div 2} = \frac{11}{2} \).
• This simplifies to \( \frac{11}{2} \), or 5.5 as a decimal.

Understanding how to simplify fractions is crucial because it reduces your final answer into an easier, more universal form. It's not only about tidying up your result but ensuring accuracy and clarity in mathematical solutions.