91Ó°ÊÓ

In linear equations, fractions can make the solving process more complicated. To eliminate them, multiply every term in the equation by the Least Common Multiple (LCM) of all denominators. For example, in the given exercise, the equation \(\frac{40 x-5}{2}+\frac{5}{2}=\frac{33-2 x}{3}-11\) involves fractions with denominators 2 and 3.
To clear these fractions, we found the LCM of 2 and 3, which is 6, and multiplied each term by 6.
This gives us:
\[6 \times \frac{40 x-5}{2} + 6 \times \frac{5}{2} = 6 \times \frac{33-2 x}{3} - 6 \times 11 \]
Simplifying, we get:
\[3(40 x-5) + 3(5) = 2(33 - 2 x) - 66 \]
This step effectively eliminates all the fractions, making the equation easier to work with.

Next, we apply the distributive property, which involves distributing a single term across terms within parentheses. This helps in breaking down and simplifying expressions.
In the equation \[3(40 x-5) + 3(5) = 2(33 - 2 x) - 66,\] we distribute 3 and 2 across the respective parentheses:
\[3 \times 40 x - 3 \times 5 + 3 \times 5 = 2 \times 33 - 2 \times 2 x - 66\]
Simplifying, this becomes:
\[120 x - 15 + 15 = 66 - 4 x - 66\]
Notices how each term is expanded out of the parentheses. This makes further simplification possible.

Combining like terms involves collecting terms with the same variable to simplify an equation further. In our example, after distributing, we get:
\[120 x - 15 + 15 = 66 - 4 x - 66\]
Combine like terms on both sides (i.e., terms that are constant and terms that include the variable x):
\[120 x = - 4 x\]
Now, to isolate x, we move all x terms to one side of the equation:
\[120 x + 4 x = 0\]
Simplifying this, we get:
\[124 x = 0\]
Finally, to solve for x, divide both sides by 124:
\[x = \frac{0}{124} \] which simplifies to:
\[x = 0\]
Always remember, combining like terms makes the equation simpler and brings us closer to finding the solution.