Solving Equations Examples And Worksheet Page 3
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Appendix A.5
Solving Equations
To solve an equation involving fractional expressions, find the least common
denominator (LCD) of all terms and multiply every term by the LCD. This process will
clear the original equation of fractions and produce a simpler equation.
An equation with a single
fraction on each side can be
Example 2
An Equation Involving Fractional Expressions
cleared of denominators by
cross multiplying. To do this,
multiply the left numerator by
x
3x
Solve
2.
the right denominator and the
3
4
right numerator by the left
Solution
denominator as follows.
x
3x
a
c
2
Write original equation.
Original equation
3
4
b
d
x
3x
ad
cb
Cross multiply.
12
12
12
2
Multiply each term by the LCD of 12.
3
4
4x
9x
24
Divide out and multiply.
13x
24
Combine like terms.
24
x
Divide each side by 13.
13
24
The solution is
x
.
Check this in the original equation.
13
Now try Exercise 23.
When multiplying or dividing an equation by a variable quantity, it is possible to
introduce an extraneous solution. An extraneous solution is one that does not satisfy
the original equation. Therefore, it is essential that you check your solutions.
Example 3
An Equation with an Extraneous Solution
1
3
6x
.
Solve
2
x
2
x
2
x
4
Solution
2
The LCD is
x
4,
or
x
2 x
2 .
Multiply each term by this LCD.
1
3
6x
x
2 x
2
x
2 x
2
x
2 x
2
Recall that the least common
2
x
2
x
2
x
4
denominator of two or more
±
fractions consists of the product
x
2
3 x
2
6x,
x
2
of all prime factors in the
x
2
3x
6
6x
denominators, with each factor
given the highest power of its
x
2
3x
6
occurrence in any denominator.
4x
8
x
2
Extraneous solution
For instance, in Example 3, by
factoring each denominator you
In the original equation,
x
2
yields a denominator of zero. So,
x
2
is an
can determine that the LCD is
extraneous solution, and the original equation has no solution.
x
2 x
2 .
Now try Exercise 35.
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