Equations involving fractions is sometimes one of the challenging problems students face. Reason being that this fractions sometimes lead to a quadratic equation while other lead to a linear equation.

Due to this they term it a “challenging problem”. However, the challenge they face here is real but thanks to God, I am here to put a light to this ever disturbing problem they encounter in this topic.

The point to be noted about this topic is that you need to follow the needed steps, if it leads to quadratic equation, you solve it using any of the methods of solving quadratic equation. If it results in a linear equation, you solve it by collection of like terms.

Having known the two most important point about equations involving fractions, let see how we can tackle, some problems.

**Solutions To Equations Involving Fractions**

**Example 1:**

Cross multiply

2x – 1 (x + 4)

Open the bracket in the right hand side

2x – 1 = x^{2} + 4x

Collect like terms

-1 = x^{2} + 4x – 2x

Take the constant to the right hand side, and equate it to zero

x^{2} + 4x – 2x + 1

Subtract the middle term

x^{2} i.e 4x – 2x = -2x

x^{2} + 4x – 2x + 1 = 0

This means the fraction has led to a quadratic equation so let solve it using factorization method

Multiply the first term with the last term i.e

The method term is 2x

Obtain two factors whose sum will give 2x and the product will yield

The factors are x and x replace the middle term 2x by the two factors x and x

Multiply the numerator

*6x/x + 21/7 = 3x ^{2}*

Divide through (Fractions)

*6+7x-6=0*

We now have a quadratic equation we multiply the first and last terms as usual

*3x ^{2} *x

*-6= -18x*

^{2}The middle term is *-7x*

The two factors are *-9x* and *2x*

So we replace *-7x *with *-9x *and *2x*

*(3x ^{2}-9x) + (2x-6) = 0*

The common terms are *3x* and 2

*3x(x-3) + 2(x-3)=0*

*(x-3) (3x+2)=0*

Separate them and equate to zero to solve for the two values of *x*

(*x-3) = 0 and (3x+2) = 0*

*x – 3 = 0 3x + 2 = 0*

*x = 3 3x/3 = -2/3*

*x = 3 or -2/3*

**Example 3**

Solve the equation *7/2x+5 -3/x+5 = 0*

Solution

The equation is *7/2x+5 – 3/x+5 = 0*

The L.C.M of the is *(2x+5) (x+5)*

Clear the fraction by multiplying each of them by the L.C.M

(*2x+5) (x+5)*

(*2x+5) (x+5)*x *7/15 – 3/x+5 *x *(2x+5)*

We now have a linear equation

*7(x+5)-3 (2x+5) = 0*

*7x+35-6x-15=0*

*7x-6x+35-15 = 0*

*x+20=0*

*x = -20*

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**Example 4:**

Solve the equation *x-5=4/x-5*

Solution

The equation is *x-5 = 4/x-5*

Multiply through by the L.C.M *x-5*

*(x-5) (x-5) = 4/x-5 *x* (x-5)*

*(x-5) (x-) = 4*

Open the bracket

*x *x *x + x *x -5* + -5 *x *x + -5x-5 = 4*

*x ^{2}-5x-5x+25=4*

*x ^{2}-10x+25=4*

Take 4 to the left hand side of the equation and equate to zero

*x ^{2}-10x+25-4=0*

*x ^{2}-10x+21=0*

The fraction has resulted in a quadratic equation, so we need to solve it by using factorization method

*x ^{2}-10x+21=0*

multiply the first and the last term

*x ^{2} *x 21

*=21x*

^{2}The middle term is *-10x*

Two factors whose sum gives *-10x *and product gives *21x ^{2}*

The two factors are *-7x and -3x*

Replace the middle term with the two factors i.e

*-10x *with *-7x *and *-3x*

*x ^{2}-7x – 3x +21=0*

*(x ^{2}-7x) –(3x+21)=0*

*x(x-7) -3 (x-7)=0*

since we have two (*x-7), *we take one

*(x-7) (x-3)= 0*

*x-7 = 0 and x-3=0*

*x=7 or x = 3*

Example 5:

Solve the equation

*½ (x-1) -1/4 (3-x) = 2*

**Solution**

The equation is

*(x-1)/2 – (3-x)/4 = 2*

Then multiply the both sides of the equation by the L.C.M of the denominator, which is 4

*4(x-1)/2 – 4(3-x)/4 = 4* x 2

*2(x-y) –(3-x) = 8*

Let open the bracket

*2x-2-3+x=8*

*2x+x-2-3= 8*

*3x – 5 = 8*

Collect like terms

*3x = 8+ 5*

*3x/3 = 13/3*

*x = 13/3*

**Conclusion**

With the solution we have provided to the problems today, answers you can tackle any problem involving equations with fractions. The various steps I laid down are enough for you to get along with problems on this topic.

Please solve simultaneously 2/e – 3/f = 1,8/e + 9/f =1/2