Hi and welcome to the Matladpi blog! In this post I will concentrate on equation solving basics.

Let us consider an equation

ax = b

To solve for x, we just need to divide both sides by the constant a, provided that a is not zero. Thus, we get

(ax) / a = b / a

(a/a)x = b / a

(1)x = b / a

x = b / a

See what we did here? By the multiplication and dividing rules,

(ax) / a = a(x / a) = x(a/a) = (a/a)x.

Careful with those calculations! For example, the calculation

(ax)/a = a / (ax)

is wrong, as it would in the first example equation lead to the following, considering that a and b are not zero:

(ax)/ a = b / a

a / (ax) = b / a

1 / x = b / a

x = a / b

Now, by inserting this wrong solution to the original equation we get

ax = b

(a*a)/b = b

a*a = b*b

This statement is not always true, as for example the statement with a = 0 and b = 1, gives

0*0 = 1*1

0 = 1

, which is not true.

As the statement a*a = b*b is not always true, the wrong solution

x = a / b is actually shown to be wrong, and thus the calculation (ax)/a = a/(ax) is wrong.

Now, by inserting the right solution x = b / a we get

ax = b

(ab)/a = b

b = b

, which is always true for every real values of a and b, provided that a is not zero.

Always check that your solution is right!

Now a little trickier one:

dx - c = 0

First, we add the constant c to the both sides, and get

dx - c + c = 0 + c

dx + 0 = c

dx = c

Seems familiar, right? We now have to just divide both sides with d, provided that d is not zero, and get

(dx) / d = c / d

(d/d)x = c / d

(1)x = c / d

x = c / d

By inserting this solution to the original equation we get

dx - c = 0

((dc)/d) - c = 0

(c) - c = 0

c - c = 0

0 = 0

True for any real values of c and d with the assumption that d is not zero.

Thus x = c / d is the right solution.

Njäf! said.

I am Jesse Sakari Hyttinen and I will see you in the next post!