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In algebra, isolating variables is a crucial step when solving linear equations. The purpose is to have the variable, in this case, \( x \), alone on one side of the equation while all the other numbers are moved to the opposite side. This process allows us to clearly see the value that the variable represents.

To isolate the variable, start by eliminating any terms that do not include the variable from the side where the variable is currently located. In the equation \(-7x + 4 = 3x\), we notice that \(-7x\) is together with the constant 4 on the left side. Our goal is to isolate the terms with \( x \) by making each side variable-centric.

To do this, add \( 7x \) to both sides of the equation. This step effectively cancels out the \(-7x\) on the left, which is added to \( 3x \) on the right, leading to \( 4 = 10x \). Achieving this form of the equation where the variable \( x \) is on one side simplifies finding its value.

Once variables are isolated, the next goal is to solve for the variable \( x \). Solving for \( x \) means determining the exact value of \( x \) that makes the equation true.

In our isolated equation, \( 4 = 10x \), \( x \) is being multiplied by 10. To solve for \( x \), you need to perform the inverse operation—division, in this case—to both sides of the equation. This method ensures that each side remains balanced.

Divide both sides by 10, which is the coefficient of \( x \). This operation looks like \( \frac{4}{10} = \frac{10x}{10} \), subsequently simplifying to \( x = \frac{4}{10} \). Now, we can see that \( x \) is equal to the fraction \( \frac{4}{10} \). However, the problem is often not fully solved until the fraction is simplified.

Simplifying fractions is an essential final step in solving algebraic equations, ensuring that your solution is in the simplest form. It involves reducing the fraction to its lowest terms without changing its value.

For the fraction \( \frac{4}{10} \), both the numerator (4) and the denominator (10) are divisible by 2. Simplifying involves dividing them each by their greatest common divisor, in this case, 2:

• Numerator: \( 4 \div 2 = 2 \)
• Denominator: \( 10 \div 2 = 5 \)

After simplification, the fraction becomes \( \frac{2}{5} \).

Therefore, the simplest form of \( x \) is \( \frac{2}{5} \). A simplified answer is easy to understand and communicate, maintaining clarity in mathematical solutions.