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 .
