- From: Adrian Blackburn
- Date: 29 August 1999
*Subject: Equation Solving*
Please can you advise me on how to go about solving this equation: |

Well, this looks to be fairly straightforward . . .

First clear the fractions by multiplying each term on both sides by (2x-1)

**3 + 4(2x-1) = 6x**

Now multiply out the brackets

**3 + 8x - 4 = 6x**

and tidy up

**8x - 1 = 6x**

Subtract 6x from both sides

**2x - 1 = 0**

add 1 to both sides

**2x = 1**

and finally divide by 2

__x = 0.5__

However, we have not finished yet. We should always check that the solution satisfies the
original equation. Replacing x by 0.5 gives

*But look! The bottom of the fraction terms works out to be zero. But dividing by zero is
not allowed! (You would get an error message on your calculator if you tried). So what has
gone wrong???*

The fact is that the solution we appear to get from the algebra is not valid. In fact, **this
equation has no solutions at all**. Why is this?

To investigate this situation, let us plot a graph of each side of the original equation:

y = 3/(2x-1) + 4

and

y = 6x/(2x-1)

We see that the two graphs never actually intersect. (If you have a graphics calculator, play
around with this yourself, zooming in and out to convince yourself.) This means that there is
no value of x for which the left hand side of the equation is equal to the right hand side.

So this is something of a "trick question", which shows the importance of checking your answers by substituting back into the original equation.

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