SOLUTION: How do you know when an equation has infinitely many solutions?

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Question 124341: How do you know when an equation has infinitely
many solutions?

Found 2 solutions by BrittanyM, bucky:
Answer by BrittanyM(80) About Me  (Show Source):
You can put this solution on YOUR website!
This will occur when your domain is not restricted.

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
If both sides of the equation can be reduced such that they are identical,
then there are
an infinite number of solutions.
.
For example, take the equation:
.
4x+=+%2812%282x+-+2%29%29%2F6+%2B+4
.
Looks pretty normal, but let's solve it. Begin by doing the distributed multiplication
in the numerator on the right side to get:
.
4x+=+%2824x+-+24%29%2F6+%2B+4
.
Divide the two terms in the numerator by the denominator 6 to reduce the equation to:
.
4x+=+4x+-+4+%2B+4
.
The -4 and +4 on the right side cancel out, so the equation is just:
.
4x+=+4x
.
and if you divide both sides by 4 you get the identity:
.
x+=+x
.
This tells you that no matter what value of x you use, the right side of the equation will
equal the left side. Try it for some values of x ...
.
Start with the equation:
.
4x+=+%2812%282x+-+2%29%29%2F6+%2B+4
.
and let x = 0
.
The equation becomes:
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0+=+%2812%280+-+2%29%29%2F6+%2B+4+=+-24%2F6+%2B+4+=+-4+%2B+4+=+0
.
The result is 0+=+0 so that works.
.
Suppose you now let x = 10. The equation:
.
4x+=+%2812%282x+-+2%29%29%2F6+%2B+4
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then becomes:
.
40+=+%2812%2820+-+2%29%29%2F6+%2B+4
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the numerator on the right side reduces to 12(18) = 216 and the equation becomes:
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40+=+216%2F6+%2B+4
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Do the division on the right side and you get:
.
40+=+36+%2B+4+
.
That is true so x = 10 works. You can try other values for x, and no matter what value you
substitute for x the result will always be the same ... the left side will equal the right
side. There are an infinite number of solutions because the equation can be reduced to
the identity x ≡ x.
.
Hope this clarifies things a little bit.
.