Equivalent linear equations

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 $2$:

$$2\cdot [5\cdot (x-1)=3x-1] \Rightarrow 10\cdot (x-1)=6x-2$$