Equation Solving

  • From: Adrian Blackburn
  • Date: 29 August 1999
  • Subject: Equation Solving

Please can you advise me on how to go about solving this equation:

Maths Help suggests:

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
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.

Return to Algebra contents list