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