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Solving Equations

We are now ready to move on to slightly more sophisticated examples.

Example

Find the solution to the equation

5( x - 3) - 7(6 - x ) = 24 - 3(8 - x ) - 3

Solution

Removing the brackets from both sides first and then simplifying:

5( x - 3) - 7(6 - x ) = 24 - 3(8 - x ) - 3

5 x - 15 - 42 + 7 x = 24 - 24 + 3 x - 3

5 x + 7 x - 15 - 42 = 3 x - 3

12 x - 57 = 3 x - 3 .

Adding 57 to both sides:

12x = 3x - 3 + 57 = 3x + 54

Subtracting 3x from both sides:

12x - 3x = 54 o r 9x = 54 giving x = 6 .

Exercise

Find the solution to each of the following equations.

(a) 2 x + 3 = 16 - (2 x - 3)

(b) 8( x - 1) + 17( x - 3) = 4(4 x - 9) + 4

(c) 15( x - 1) + 4( x + 3) = 2(7 + x )

Solutions

(a) 2 x + 3 = 16 - (2 x - 3) = 16 - 2 x + 3 = 19 - 2 x

Now add 2x to both sides and subtract 3 from both sides

2 x + 3 = 19 - 2 x

4 x + 3 = 19

4 x = 19 - 3

4 x = 16

and the solution is x = 4 . This can be checked by putting x = 4 in both sides of the first equation above and noting that each side will have the value 11.

(b)

8( x - 1) + 17( x - 3) = 4(4 x - 9) + 4

8 x - 8 + 17 x - 51 = 16 x - 36 + 4

25 x - 59 = 16 x - 32

25 x - 16 x - 59 = - 32

9 x - 59 = - 32

9 x = 59 - 32

9 x = 27

x = 3 .

Inserting x = 3 into the equation we can check that both sides have the value 16.

(c)

15( x - 1) + 4( x + 3) = 2(7 + x )

15 x - 15 + 4 x + 12 = 14 + 2 x

19 x - 3 = 2 x + 14

19 x - 2 x - 3 = 14

17 x - 3 = 14

17 x = 14 + 3 = 17

x = 1

Inserting x = 1 into the equation we can check that both sides have the value 16.

Quiz

Which of the following is the solution to the equation

5x - (4x - 7)(3x - 5) = 6 - 3(4x - 9)(x - 1) ?

(a) - 2 (b) - 1 (c) 2 (d) 4

Solution:

First expand the brackets separately using FOIL (see the package on Brackets ):

These can now be substituted, carefully, into the equation:

5 x - [ (4 x - 7)(3 x - 5) ] = 6 - 3 [ (4 x - 9)( x - 1) ]

5 x - [ 12 x - 41 x + 35 ] = 6 - [ 4 x - 13 x + 9 ]

5 x - 12 x + 41 x - 35 = 6 - 4 x + 13 x - 9

- 12 x + 46 x - 35 = - 12 x + 39 x - 21 .

Notice the extra pair of square brackets in the first equation above. These are to emphasise that the negative sign multiplies all of the parts inside the [ ] brackets. The procedure now follows in an obvious manner. Add 12 x to both sides, subtract 39 x from both sides then add 35 to both sides:

- 12 x + 46 x - 35 = - 12 x + 39 x - 21

46 x - 35 = 39 x - 21

46 x - 39 x - 35 = - 21

46 x - 39 x = 35 - 21

7 x = 14

x = 2 .

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