91Ó°ÊÓ

When solving linear equations, the first step is to isolate the variable. This means you want to get the variable by itself on one side of the equation.
For instance, in the equation \(-\frac{5}{7} x - 1 = 3\), we start by eliminating any constants on the side of the equation with the variable.
You do this by performing the opposite operation.
Here, we add 1 to both sides because \(-1 \) is subtracted from the variable term.
Doing so gives us: \(-\frac{5}{7}x - 1 + 1 = 3 + 1\)
. This simplifies to \(-\frac{5}{7}x = 4\).
Now, the variable term is isolated, meaning it's by itself on one side of the equation.

The next step involves getting rid of the fraction that the variable is multiplied by. This is where reciprocals come in handy.
A reciprocal of a fraction is created by swapping the numerator and the denominator.
For example, the reciprocal of \(-\frac{5}{7}\) is \(-\frac{7}{5}\). Multiplying by the reciprocal cancels out the fraction.
In our equation, we have \(-\frac{5}{7}x = 4\), so we multiply both sides by \(-\frac{7}{5}\) to eliminate the fraction:

\(-\frac{7}{5} \times -\frac{5}{7} x = -\frac{7}{5} \times 4\)

By multiplying the left side, \(-\frac{7}{5} \times -\frac{5}{7}\), we get 1 because the fractions cancel each other out, leaving us with \(\text{x}\).
Thus, it simplifies to \(\text{x} = -\frac{28}{5}\).

Dealing with negative fractions is an important skill in solving linear equations.
When a fraction is negative, it means the numerator, the denominator, or both, contain a negative sign.
For example, in \(-\frac{5}{7}\), the negative sign can be thought of as applying to the whole fraction.

In our problem, we moved the negative fraction \(-\frac{5}{7}\) by multiplying both sides of the equation by its reciprocal, which also has a negative sign.
When multiplying two negative numbers, the result is positive.
So, \(-\frac{7}{5} \times -4\) equals \(\frac{28}{5}\).
Finally, the variable is solved: \(\text{x} = -\frac{28}{5}\).
Working carefully with negative fractions ensures accuracy in your solution.