91Ó°ÊÓ

Isolating the variable in an algebraic equation is a foundational technique for solving for the unknown. This involves rearranging the equation so that the variable we are trying to find is on one side of the equation by itself. In our exercise, we begin with \(\frac{2}{5} y + 4 = 6\). To isolate \(y\), we need to get rid of the constant term that is on the same side as \(y\), which is the +4 in this case. To do so, we subtract 4 from both sides of the equation:
\[\frac{2}{5} y + 4 - 4 = 6 - 4\]
Which simplifies to:
\[\frac{2}{5} y = 2\].
Now, \(y\) is the only term on one side of the equation, effectively 'isolated', ready for the next step to clear the fraction and solve for \(y\).

When an equation involves fractions, it can make the process of finding the variable seem more complicated. To simplify the process, one approach is to clear the fractions, which we achieve by multiplying every term by the least common denominator. In our example, however, we can simply multiply both sides by the reciprocal of the coefficient of \(y\). The reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\). When we multiply both sides by this reciprocal, the fraction on the left side cancels out:
\[\frac{5}{2} \cdot \frac{2}{5} y = \frac{5}{2} \cdot 2\]
This simplifies to:
\[y = \frac{5}{2} \cdot 2\].
By performing this multiplication, the variable is free from fractions and can be easily solved. After completing the calculation on the right side, we find that \(y = 5\).

After finding a potential solution to an equation, it's crucial to verify its correctness by substituting it back into the original equation. This step checks if the equation balances with the proposed solution. In our case, the proposed solution is \(y = 5\). To check, we substitute \(y\) with 5 in the original equation:
\[\frac{2}{5} \cdot 5 + 4 = 6\]
Simplify the left side of the equation: \[\frac{2}{5} \cdot 5 = 2\] and then \[2 + 4 = 6\]. However, the simplification was mistakenly made here, as \(\frac{2}{5} * 5\) actually equals 2, and thus the correct simplification should be \[2 + 4 = 6\], which indeed simplifies to \[6 = 6\].
If the equation does not simplify correctly, as in our revised step, it means the solution is valid. However, if the original and final expressions are not equal, then the solution would have to be reassessed.