# Equations with fractions

**Solving equations with fractional coefficients**

Equations which have fractional coefficients will still follow the same rules of fractions when working them out.

For example, when solving the following equation

– 3 = 7 – *x*

= 8(* – *3) = 8(7 – *x*) (remove the denominator by multiplying both sides by 8)

= 6*x – *3 = 56 – 8*x* (expand the brackets)

= 14*x – *3 = 56 * (*add 8* x *to both sides)

= 14*x – *3 = 56 (add 3 to both sides)

= 14*x* = 59 (divide both sides by 14)

= *x* = 4.21 (to 2 decimal places)

Points to note:

- can also be written as
- If an equation contains more than one fraction, these can be removed by multiplying by the lowest common denominator

For example

= *x* = *x* + 1 (lowest common denominator is 4)

= 4( *x* = 2( *x* + 1) (expand the brackets)

= 3*x* = *x* + 2 (subtract *x* from both sides)

= 2*x* = 2 (divide by 2 on both sides)

= *x* = 1

**Solving equations involving division**

Solving equations involving division uses similar principals as with fractions. For example:

= = *x* – 3 (remove the denominator from both sides by multiplying by 5)

= 2*x* + 8 = 5(*x* – 3) (expand the brackets)

= 2*x* + 8 = 5*x* – 15 (add 15 to both sides)

= 2*x* + 23 = 5*x* (subtract 2*x* from both sides)

= 23 = 3*x (*divide both sides by 3)

= *x* = 7.67 (to 2 decimal places)

**Solving equations involving dual fractions**

If both sides of an equation are divided by a number, these numbers are removed by multiplying both sides by the lowest common denominator. For example,

(lowest common multiple of 4 & 3 is 12, multiple both sides)

=

= (simply and remove the denominator, by dividing the numerator by the denominator)

= 3(3*x* – 2) = 4(3*x* – 1) (expande the brackets)

= 9*x* – 6 = 12*x* – 4 (add 6 to both sides)

= 9*x* = 12*x + 2 *(subtract 12*x* from both sides)

= -3*x = 2 *(divide both sides by -3)

= *x =
*

**Solving equations using cross multiplication**

In the example above, the equation could also be solved using cross multiplication and then it is solved as usual. This is a more efficient way of re-arranging an equation and an example of cross multiplication is as follows:

The denominator is removed by multiplying the first side by 3 and the second side by 4.

Therefore

= 3(6*x* – 2) = 4(3*x* – 1)

The equation is then solved using the normal method.

**Solving equations involving division of unknowns**

There are also times when there are unknowns as a denominator. In this situation, the same rules are applied as cross multiplication.

For example,

Similar to cross multiplication, the denominators on the left side is multiplied to the right side and the denominators on the right side is multiplied to the left side resulting in the following:

5(2*x* – 5) = 6(*x* + 4)

The equation is then solved using the normal method.