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III SOLVING LINEAR EQUATIONS
A linear equation is an equation of the form ax + b = 0. The degree of a linear
equation is one. A linear equation may have no solution, one solution or an infinite
number of solutions.
To solve a linear equation :
1) Simplify each side of the equation
2) If there are any indicated additions or subtractions, use the inverse operations
to undo these.
3) If there are any indicated multiplication or division involving the variable,
use the inverse operation to undo these.
1
EXAMPLE 1. Solve m  6  (8 18m)
2
m  6  4  9m
m  6  9m  4  9m  9m
10m  6  4
10m  6  6  4  6
10m  10
10m 10

10 10
m 1
The solution m = 1 should be checked by replacing m with 1 in the
original equation.
1
1  6  (8 18 1)
2
1
 5   8 18 
2
1
 5  (10)
2
 5  5
The check of the solution can be done mentally.
EXAMPLE 2. Solve the equation : 2(y - 8) = 3y - 10 - (y + 6)
2( y  8)  3 y 10  ( y  6)
2 y 16  3 y 10  y  6
2 y 16  2 y 16
When the equation is identical on each side, the equation can be satisfied
by every real number. This kind of equation is called an identity equation. The solution
to an identity equation is the set of real numbers, and is designated as .
EXAMPLE 3. Solve the equation : 4w - 7 = 2(2w - 1)
4w  7  2(2 w 1)
4w  7  4w  2
4w  7  4w  4w  2  4w
7   2
Since the equation is equivalent to the false statement " -7 = -2", the
equation is said to have no solution. The solution set is the empty set or the null set,
and is designated by  .
EXAMPLE 4. Solve the equation : 5x - 2(1 - x) = 2(2x - 1)
5 x  2 1  x   2(2 x  1)
5x  2  2 x  4 x  2
7x  2  4x  2
7x  4x  2  4x  2  4x
3x  2   2
3x  2  2   2  2
3x  0
3x 0

3 3
x0