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!