Download How many solutions does the equation 3(x − (5 − 2x) + 42) = 2 − 6x

How many solutions does the equation
3(x − (5 − 2x) + 42) = 2 − 6x
have? The only operations used to build up either side are multiplying by a constant and addition. So both sides have to simplify
down to a linear expression, of the form ax + b where a and b are
constants.
The equation simplifies down to
ax = b.
If a 6= 0, this equation simplifies to
b
x= .
a
In this case a = 15. How could we know have known in advance
that a 6= 0?
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If a = 0 how many solutions does it have? If a = 0 and b = 0,
then the equation simplifies to
0 = 0.
This is has infinitely many solutions because for all x, 0 = 0.
If a = 0 and b 6= 0, then the equation has no solutions. The
equation simplifies to
0=b
Divide both sides by b since b 6= 0 and get
0 = 1.
This is not true for any value of x.
2
So, we know that if
3(x − (5 − 2x) + 42) = 2 − 6x
then
−109
x=
.
15
Is this the same as saying if
x=
−109
15
then
3(x − (5 − 2x) + 42) = 2 − 6x?
No. So to finish solving this equation, we need to substitute
x = −109/15 into the original equation and check that it is a
solution.
3
You might think it was silly to check the solution in the last
example, but look at the following example. If
√
x = −5
then
x = 25.
Now here we really need to check; in fact x = 25 is not a solution
of the original equation; the equation has no solutions. The ifthen statement is like saying “If you are the President of the US
then I’m the Pope.” Neither the if part nor the then part is true,
although the if-then statement itself is true.
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The difference between the two situations—the linear equation
we started with and this one with the square root—is that in
the first on we solve by adding and subtracting and multiplying
or dividing by a nonzero constant. These are all invertible (i.e.,
reversible) operations. In the second example we have to square
both sides, which is not an invertible operation.
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Suppose you solve an equation by a series of steps like:
If bla = bla then . . . [lots of steps] . . . x = number.
At each stage you either simplify one of the sides, or you apply
a function to both sides. In the case of linear equation, the only
functions needed are f (x) = x + a and f (x) = xb, b 6= 0. These
are invertible: you can undo these functions (by subtracting a
or dividing by b). If all the operations in solving an equation are
invertible then you can reverse all of the implications and say
If x = number then . . . [reverse all the steps] . . . bla = bla.
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