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Reducing Equations to Simpler Form

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Reducing Equations to Simpler Form:

When linear equations are in fractions then we can reduce them to a simpler form by -

  • Taking the LCM of the denominator

  • Multiply the LCM on both sides, so that the number will reduce without the denominator and we can solve them by the above methods.

Example

Solve: `(6x + 1)/3 + 1 = (x - 3)/6`

Multiplying both sides of the equation by 6, 

`(6(6x + 1))/3 + 6 xx 1 = (6(x - 3))/6`
or 2(6x + 1) + 6 = x – 3  
or 12x + 2 + 6 = x – 3         ......(opening the brackets ) 
or 12x + 8 = x – 3 
or 12x – x + 8 = – 3 
or 11x + 8 = – 3 
or 11x = –3 – 8 
or 11x = –11 
or x = – 1

Check:  
LHS = `(6(-1) + 1) / 3 +1 = (-6+1)/3 + 1 = (-5)/3 + 3/3 = (-5+3)/3 = (-2)/3`.

RHS = `((-1)-3) /6 = (-4)/ 6 = (-2)/3`.

LHS = RHS     .......(as required)

Example

Solve: 5x - 2(2x - 7) = 2(3x - 1) + `7/2`

Let us open the brackets,

`"LHS" = 5x - 4x + 14 = x + 14`.

`"RHS" = 6x - 2 + 7/2 = 6x - 4/2 + 7/2 = 6x + 3/2`.

The equation is `x + 14 = 6x + 3/2`

or  `14 = 6x - x + 3/2`

or  `14 = 5x + 3/2`

or  `14 - 3/2 = 5x`

or  `(28 - 3)/2 = 5x`

or  `25/2 = 5x`

or `x = 25/2 xx 1/5 = (5 xx 5)/(2 xx 5) = 5/2`.

Therefore, required solution is `x = 5/2`

Check:

`"LHS" = 5 xx 5/2 - 2(5/2 xx 2 - 7) = 25/2 - 2(5 - 7) = 25/2 - 2(- 2) = 25/2 + 4 = (25 +8)/2 = 33/2.`

`"RHS" = 2(5/2 xx 3 - 1) + 7/2 = 2(15/2 - 2/2) + 7/2 = (2 xx 13)/2 + 7/2 = (26 + 7)/2 = 33/2.`

LHS = RHS.    ....(As required)

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Shaalaa.com | Problems on Linear Equations in One Variable

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