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# EQUATIONSWITH FRACTIONS

Clearing of fractions

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.

 x3 + 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 both sides of the equation by 15.

 15· x3 + 15· x − 2   5 = 15· 6

On the left, distribute 15 to each term. Each denominator will now divide into 15 -- that is the point -- 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. (Lesson 1.) 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:

 x2 − 5x 6 = 19

Solution.   The LCM of 2, 6, and 9  is 18. (Lesson 23 of Arithmetic.) Multiply both sides by 18 -- and cancel.

9x − 15x  =  2.

It should not be necessary to actually write 18.  The student should simply look at and see that 2 will go into 18 nine (9) times. That term therefore becomes 9x.

Next, look at , and see that 6 will to into 18 three (3) times.  That term therefore becomes 3· −5x = −15x.

Finally, look at , 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 = − 13

Example 3.   Solve for x:

½(5x − 2) = 2x + 4.

Solution.   This is an equation with a fraction. Clear of fractions by mutiplying both sides by 2:

 5x − 2 = 4x + 8 5x − 4x = 8 + 2 x = 10.

In the following problems, clear of fractions and solve for x:

To see each answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

 Problem 1. x2 − x5 = 3 The LCM is 10.  Here is the cleared equation and its solution: 5x − 2x = 30 3x = 30 x = 10.

On solving any equation with fractions, the very next line you write --

5x − 2x  =  30

-- should have no fractions.

 Problem 2. x6 = 1 12 + x8 The LCM is 24.  Here is the cleared equation and its solution: 4x = 2 + 3x 4x − 3x = 2 x = 2
 Problem 3. x − 25 + x3 = x2 The LCM is 30.  Here is the cleared equation and its solution: 6(x − 2) + 10x = 15x 6x − 12 + 10x = 15x 16x − 15x = 12 x = 12.

Problem 4.  A fraction equal to a fraction.

 x − 14 = x7 The LCM is 28.  Here is the cleared equation and its solution: 7(x − 1) = 4x 7x − 7 = 4x 7x − 4x = 7 3x = 7 x = 73

We see that when a single fraction is equal to a single fraction, then the equation can be cleared by "cross-multiplying."

 If ab = cd , then ad = bc.
 Problem 5. x − 33 = x − 5   2 Here is the cleared equation and its solution: 2(x − 3) = 3(x − 5) 2x − 6 = 3x − 15 2x − 3x = − 15 + 6 −x = −9 x = 9
 Problem 6. x − 3x − 1 = x + 1x + 2 Here is the cleared equation and its solution: (x − 3)(x + 2) = (x − 1)(x + 1) x² −x − 6 = x² − 1 −x = −1 + 6 −x = 5 x = −5.
 Problem 7. 2x − 39 + x + 1   2 = x − 4 The LCM is 18.  Here is the cleared equation and its solution: 4x − 6 + 9x + 9 = 18x − 72 13x + 3 = 18x − 72 13x − 18x = − 72 − 3 −5x = −75 x = 15.
 Problem 8. 2x − 3 8x = 14 The LCM is 8x.  Here is the cleared equation and its solution: 16 − 3 = 2x 2x = 13 x = 13 2

2nd Level Next Lesson:  Word problems

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