Vocabulary
Numerator - The number of expression written above the line in a fraction.
Denominator - The quantity below the line in a fraction, telling the number of equal parts the whole is
divided into.
Common Denominator - The same number on the bottom of all fractions involved in the equation.
Simplify - Combine like terms and apply properties to an expression to make computation easier.
Equivalent - Naming the same number.
TO SOLVE AN EQUATION WITH fractions, we transform it into an equation without fractions -- which we know how to solve. The technique is called clearing of fractions.
Example 1. Solve for x:
| x 3 |
+ | x − 2 5 |
= 6. |
Solution. Clear of fractions as follows:
Multiply both sides of the equation -- every term -- by the LCM of denominators. Each denominator will then divide into its multiple. We will then have an equation without fractions.
The LCM of 3 and 5 is 15. Therefore, multiply every term on both sides by 15:
| 15· | x 3 |
+ | 15· | x − 2 5 |
= 15· 6 |
Each denominator will now divide into 15 -- that is the point
-- and we have the following simple equation that has been "cleared" of fractions:
-- and we have the following simple equation that has been "cleared" of fractions:| 5x + 3(x − 2) | = | 90. |
| It is easily solved as follows: | ||
| 5x + 3x − 6 | = | 90 |
| 8x | = | 90 + 6 |
| x | = | 96 8 |
| = | 12. | |
We say "multiply" both sides of the equation, yet we take advantage of the fact that the order in which we multiply or divide does not matter. Therefore we divide the LCM by each denominator first, and in that way clear of fractions.
(We choose a multiple of each denominator, because each denominator will then be a divisor of it.)
Example 2. Clear of fractions and solve for x:
| x 2 |
− | 5x 6 |
= | 1 9 |
Solution. The LCM of 2, 6, and 9 is 18. Multiply each term by 18 -- and cancel.
9x − 15x = 2.
It should not be necessary to actually write 18. The student should
| simply look at | x 2 |
, and see that 2 will go into 18 nine (9) times. That term |
therefore becomes 9x.
| Next, look at | 5x 6 |
, and see that 6 will to into 18 three (3) times. That |
term therefore becomes 3· −5x = −15x.
| Finally, look at | 1 9 |
, and see that 9 will to into 18 two (2) times. That |
term therefore becomes 2 · 1 = 2.
Here is the cleared equation, followed by its solution:| 9x − 15x | = | 2 | |
| −6x | = | 2 | |
| x | = | 2 −6 |
|
| x | = | − | 1 3 |
Example 3. Solve for x:
½(5x − 2) = 2x + 4.
Solution. This is an equation with a fraction. Clear of fractions by mutiplying every term by 2:
| 5x − 2 | = | 4x + 8 |
| 5x − 4x | = | 8 + 2 |
| x | = | 10. |
In the following problems, clear of fractions and solve for x:
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Do the problem yourself first!
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Do the problem yourself first!
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