Rule of Changing Signs in an Equation

The rule of changing signs states that if we reverse the signs of both sides of an equation, we obtain an equivalent equation. $$ a - b = c - d \ \ \ \Leftrightarrow \ \ \ b-a = d-c $$

For example, consider the equation:

$$ 3x - 5 = -2 $$

To apply the rule of sign change, multiply both sides by -1.

$$ (3x - 5) \cdot (-1) = -2 \cdot (-1) $$

This produces an equivalent equation to the original one, but with every term having the opposite sign.

$$ -3x + 5 = 2 $$

Note. According to the principles of equation equivalence, multiplying both sides of an equation by the same nonzero number always yields an equivalent equation.

Why change the sign of an equation?

The rule of sign change is often useful when simplifying or solving an equation.

In fact, there’s no inherent preference for having an equation written with positive or negative terms - both forms are mathematically equivalent.

However, in some situations, one form may be more convenient than the other.

For instance, when isolating a variable, it usually feels more natural to express it with a positive sign.

Example. Consider the equation $$ -2x = 8 $$. To solve it, divide both sides by two: $$ \frac{-2}{2} = \frac{8}{2} $$ $$ -x = 4 $$ The result -x = 4 is correct, but it’s often clearer to rewrite it so that the variable appears with a positive sign: $$ x = -4 $$

When adding or subtracting two equations, it’s also simpler to work with the version where all terms are positive, avoiding unnecessary sign changes during calculations.

For example, it’s generally easier to compute 102 - 88 than -88 + 102 in your head.

Example. Take the equation $$ (-2x-4) - (-2x - 4) = -4x $$ It’s often neater to rewrite it in the equivalent form $$ (2x+4) + (2x-4) = 4x $$ This eliminates the need to flip signs while rearranging terms, making the arithmetic smoother. The equation also looks cleaner, since the leading “+” sign is implied and can be omitted from the expressions.

Ultimately, the choice of which form to use depends on context and personal preference.

The important thing to remember is that both versions are completely equivalent - you can always move from one to the other by applying the rule of changing signs.

And so on.