The equation:
$$$x-2=3$$$
has the solution:
$$$x=3+2 \Rightarrow x=5$$$
While in this second equation:
$$$3x-3=2x+2$$$
the solution is:
$$$3x-2x=2+3 \Rightarrow x=5$$$
When two equations have the same solution it is said that they are equivalent equations.
There are a couple of basic rules to generate equivalent equations:
- When we add or subtract the same number on both members of an equation an equivalent equation is obtained.
In the first example, if we add $$3$$ on both sides of the equality, we obtain:
$$$x-2+3=3+3 \Rightarrow x+1=6$$$
This equation is completely equivalent to the first one. It is possible to verify it by checking that they have the same result:
$$$x+1=6 \Rightarrow x=6-1 \Rightarrow x=5$$$
- If we multiply or divide both members of the equation by the same number, an equivalent equation is obtained.
For instance, if we multiply both sides of the first equation by $$2$$, we obtain:
$$$2(x-2)=2(3)\Rightarrow 2x-4=6$$$
The obtained equation is equivalent to the first one. It is verified by solving it:
$$$2x=6+4 \rightarrow 2x=10 \Rightarrow x=\frac{10}{2}=5$$$
The latter point is interesting in order to eliminate denominators of the equations, so they are simplified, thereby making them easier to solve.
In the following equation:
$$$\displaystyle -5-\frac{x}{3}=11$$$
If we multiply by $$3$$, the denominator is eliminated:
$$$\displaystyle 3\Big(-5-\frac{x}{3}=11\Big) \Rightarrow -15-x=33$$$
This second equation is equivalent to the first one and it is very easy to solve:$$$-x=33+15 \Rightarrow -x=48 \Rightarrow x=-48$$$
A certain agility to generate equivalent equations is useful when creating exercices. The starting point for raising an equation is to know its result in advance.
For instance, if we want $$x=2$$, the following equation is a possibility:
$$$2x-5=-1$$$
Since if we replace the result the equality is supported:
$$$2 \cdot 2 -5 =-1 \Rightarrow 4-5=-1 \Rightarrow -1=-1$$$
Now we can generate an equivalent equation to make the equation seem more complicated. For example, we can write $$-5$$ as the expression $$-3-2$$ and move their position:
$$$-3+2x-2=-1$$$
We can also break down the unknown. For example: we can express $$2x$$ as $$5x-3x$$, but moving $$-3x$$ to the other side of the equality, with its change in the sign:
$$$-3+5x-2=-1+3x$$$
Now, operating the first member, we get:
$$$5x-5=3x-1$$$
In this case it is possible to extract common factor for the first member (5), so we can introduce brackets:
$$$5(x-1)=3x-1$$$
Finally, we can multiply the whole equation by the same number, for example $$2$$:
$$$2\cdot [5\cdot (x-1)=3x-1] \Rightarrow 10\cdot (x-1)=6x-2$$$