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Solve Linear Equations with Fractions
Let's review those 5 steps to solve linear equations:
- Get rid of parentheses by distributive property.
- Combine like terms.
- Move variable terms to one side of the equal sign.
- Move number terms to the other side of the equal sign.
- Get rid of the number in front of the variable.
If there are fractions in an equation, like 9
3
2 x โ = , it would be great if we can avoid fraction addition
and division. Good news! I will show you a trick to get rid of fractions in an equation. The trick's name is
Fraction Buster. First, we need to learn a property of linear equations.
Multiply Each Term by the Same Number
If we have an equation, say 1 + 2 = 3 , and if we multiply each term with the same number, say 2 , will
the equation still be true? Let's see:
It is true! If we have an equation, we can multiply each term with the same number, and the equation
will still be true.
Now we can solve 9
3
2 x โ =. To get rid of 3
, we multiply each term by 3 :
x
x
x
Note that each term must be multiplied by 3 , including 2 x and 9. Next, we follow those 5 steps:
x
x
x
x
Note that although we got rid of fractions in the equation to make our life easier, the solution could still
be a fraction.
[ Example 1 ] Solve
5
p for p.
[ Solution ] To get rid of the fractions, we multiply each term by 5 :
p
p
p
Again, it's a common mistake not to multiply 2 with 5 and get 4 p โ 2 = 6. EACH term in the equation
must be multiplied by the same number.
The rest is easy:
p
p
p
Common Denominator
What if we have two fractions with different denominators?
[ Example 3 ] Solve
10
q for q.
[ Solution ] This time, if we multiply each term by 5, the fraction
10
will not go away, because
5 โ
= โ
= =. We need to find a number both 5 and 10 go into. Recall the concept of
common denominator? The common denominator of
5
4 q and 10
is 10, so we multiply each term by
10, and we have:
Invisible Parentheses
One more new thing to learn is about "invisible parentheses". We know 4
3
6 โ
=. If we re-write 3
into
, then 4 3
โ
should still be true. Let's verify:
6 = รท โ
+ = โ
+ = + =
Oops! What went wrong?
This is because there is an invisible pair of parentheses around 1 + 1 , so
3
remains to be 3
. We
should have done:
6 = รท โ
+ = รท โ
= โ
=
Following this logic, we have:
6 = รท โ
+ = โ
+ = +
โ
x x x
x
The lesson is: When you use Fraction Buster, if you have more than one term in the numerator, add a
pair of parentheses around those terms.
[ Example 5 ] Solve
4
x for x.
[ Solution ] The common denominator of those two fractions is 12, so we multiply each term with 12:
x
x
x
x
x
x
x
To summarize what we learned in this lesson: When we solve a linear equation, if there are fractions,
multiple each term with the common denominator to get rid of all fractions, and then go through those
5 steps to solve a fraction-free linear equation.
