If we multiply through by 8, we get
(5/2)× 8 = (20/8) × 8
which simplifies to 40/2 = 160/8, and further to 20 = 20.
Dividing through by a quantity
We can divide any equation through by a nonzero number, and we’ll always get another
valid equation. Dividing by any nonzero number is the same thing as multiplying by its
reciprocal.
As we’ve already learned, when we divide an equation through by some complicated
expression that contains one or more variables, we must be careful! If that expression can
become equal to 0, we’re in trouble. The danger is worsened by the fact that it can be difficult
to tell when such trouble is taking place—until it’s too late.
As an example of what can happen when we get careless about this, suppose we come
across this equation and are told to find the value of x:
x^2 +x+ 3 = 3We might start by subtracting 3 from each side. That is perfectly legitimate. We then get
x^2 +x= 0Remembering that any quantity squared is equal to that quantity multiplied by itself, we
divide the equation through by x, getting
x^2 /x+x/x= 0 /xBecause x/x= 1 and 0 /x= 0 no matter what x might happen to be (or so we think in the
excitement of the moment), we can simplify this to
x= 0Confident that we have solved the equation, we “plug in” 0 for x in the original and check
it out:
02 + 0 + 3 = 3
This simplifies to 3 = 3. “The problem has been solved,” we say.
Not so fast! We’ve missed the other solution, which is x=−1. Check it out. Try “plugging
in” −1 for x in the original equation, and see what happens. This oversight occurred because
we made the mistake of dividing through by x when one of the two solutions to the original
equation is x= 0. This blinded us to the existence of the other solution.
Equation Morphing Revisited 175