91Ó°ÊÓ

When solving an algebraic equation like the given one, our goal is to solve for the variable, in this case, `x`. A common first step in achieving this is to isolate the variable on one side of the equation. This means we need to move other numbers or terms away from the variable.
For the equation provided: \[3x + \frac{2}{5} = \frac{1}{10}\] we need to isolate `3x` to then solve for `x`. To do this, we remove the fraction \(\frac{2}{5}\) that is added to `3x` by performing the opposite operation, which is subtraction, on both sides of the equation. Subtracting \(\frac{2}{5}\) from both sides results in:\[3x + \frac{2}{5} - \frac{2}{5} = \frac{1}{10} - \frac{2}{5}\] This simplifies to: \[3x = \frac{1}{10} - \frac{2}{5}\] By isolating `3x`, we have completed the first step, setting up for further simplification.

Once the term with the variable is isolated, it is often necessary to simplify fractions to continue solving the equation. Simplifying involves performing operations such as addition or subtraction by making sure fractions have a common denominator.
In the equation: \[3x = \frac{1}{10} - \frac{2}{5}\] we see subtracting \(\frac{2}{5}\) from \(\frac{1}{10}\). Since the denominators are different (10 and 5), we need to make them the same to simplify.

• The least common denominator between 10 and 5 is 10.
• Convert \(\frac{2}{5}\) to \(\frac{4}{10}\).

Now, subtract the fractions: \[\frac{1}{10} - \frac{4}{10} = -\frac{3}{10}\] Thus, the simplified equation becomes: \[3x = -\frac{3}{10}\] By simplifying, we create a cleaner equation for the final steps.

Finally, we focus on solving for the variable, which means finding its value. After isolating and simplifying, we're ready for this step. The equation after simplification is:\[3x = -\frac{3}{10}\] To solve for `x`, divide both sides by 3 to undo the multiplication of `x` by 3. Dividing gives:\[x = \frac{-\frac{3}{10}}{3}\] This further simplifies by multiplying the numerator -\(\frac{3}{10}\) by the reciprocal of 3, which is \(\frac{1}{3}\):

• \(x = -\frac{3 \times 1}{10 \times 3}\)
• \(x = -\frac{3}{30}\)

Simplify \(-\frac{3}{30}\) by finding the greatest common factor of 3 and 30, which is 3. Divide both the numerator and the denominator by 3:\[x = -\frac{1}{10}\] Finding the value of `x` shows the process of solving an algebraic equation step by step.