Algebra Demystified 2nd Ed

Chapter 7 linear equaTionS 211

Anytime we multiply (or divide) both sides of the equation by an expression
with a variable in it, we must check our solution(s) in the original equation.
When we cross-multiply, we are implicitly multiplying both sides of the equa-
tions by the denominators of each fraction, so we must check our solution in
this case as well. The reason that we must check our solution is that a solution
to the converted equation might cause a zero to be in a denominator of the
original equation. Such solutions are called extraneous solutions. Let us see what
happens in the next example.

EXAMPLES

1

2

3

2

6

22

22

xx

x

x

xx

−

=

+

−

−+

−−

()()

TheLCDis()().

()()^1 ()()

()()

xx

x

xx

x

x

x

−+

−

=− +

+

−

−+









22

2

223

2

6

22 x 

Multiply each side by the LCD.

xxx

x

xx x

xx

+= −+

+

−− +

−+

222 3

2

226

22

()(()( )

()()

)

Distribute the LCD.

xxx

xxx

xx

xx

x

+= −−

+= −−

+=−−

++

+=−

−

23 26

23 66

236

33

426

()

222

48

2

−

=−

=−

x

x

But x = −2 leads to a zero in a denominator of the original equation, so

x = −2 is not a solution to the original equation. The original equation, then,

has no solution.

EXAMPLES EXAMPLES