SOLVE LINEAR EQUATIONS
A linear equation is one in which the variables are to the first power;
ax + b = c (a, b, c are constants)
Examples are 2x - 5 = 4 and 3x + 5y = 12.
When you are asked to solve a linear equation you are being asked to find the value of the
variable which will make the equation a true statement.
Example:
x = 4 is the solution to the equation 2x - 4 = 4. That means that if I substitute 4 into the equation
I will get a true statement; 2(4) - 4 = 8 - 4 = 4.
Properties of Equality allow you to do the steps necessary to solve an equation.
For all real numbers a, b, and c:
1) Addition Property:
2) Multiplication Property:
if a = b, then a ± c = b ± c
if a = b, then ac = bc
The properties state that you can add/subtract/multiply/divide the same number to both sides of
the equation and keep the equality.
KEEP IN MIND:
1) The object in solving an equation is to isolate the variable to one side with a coefficient of
positive one.
2) If you are bringing a term to the opposite side of the equal symbol, then do the opposite
operation.
GUIDELINE:
1)
2)
2)
3)
4)
5)
6)
Simplify any grouping symbols.
If you would like to eliminate fractions, then multiply each term by the LCD.
Combine like terms.
Bring variables to one side of equal symbol.
Apply the addition property.
Apply the multiplication/division property.
Check your solution in the original equation.
EXAMPLES:
1) Solve x + 7 = 29
x = 22 is the solution.
subtract 7 on both sides
2) Solve 8 = a – 32.
a = 40 is the solution.
add 32 on both sides
3) Solve
3
+ m = - 12
4
3
m =  12 is the solution.
4
subtract 3/4 on both sides
4) Solve 6x = -90
x = -15 is the solution.
divide both sides by 6
5) Solve x = 12
5
x = 60 is the solution.
multiply both sides by 5
6) Solve 2x = 8
3
x = 12 is the solution.
divide both sides by 2/3
or
multiply both sides by the
reciprocal, 3/2
7) Solve 4x – 3 = 29
4x = 32
x = 8 is the solution.
add 3 to both sides
divide both sides by 4
8) Solve 7x + 12 = 13x - 21.
-6x + 12 = -21
-6x = -33
x = 33/6 is the solution.
subtract 13x on both sides
subtract 12 on both sides
divide both sides by -6
NOTE:
1) The solution can be any real number. Therefore, 33/6 is equivalent to 5 ½ or 5.5.
2) There is more than way to solve equations. However, any correct approach will yield the
same solution. Remember what it means to find the solution.
For example: Solve 7x + 12 = 13x - 21 can be done differently.
12 = 6x - 21
subtract 7x on both sides
33 = 6x
add 21 on both sides
33/6 = x is the solution.
divide both sides by 6
9) Solve 3(2x - 1) = 4(x + 5).
6x - 3 = 4x + 20
simplify by distributing
2x - 3 = 20
subtract 4x on both sides
2x = 23
add 3 on both sides
x = 23/2 or 11.5 is the solution. divide both sides by 2
10) Solve 8 - x - (-12) = 14 + 3x.
8 - x + 12 = 14 + 3x
20 - x = 14 + 3x
20 = 14 + 4x
6 = 4x
6/4 or 1.5 = x is the solution.
simplify grouping symbol
collect like terms
add x on both sides
subtract 14 on both sides
divide both sides by 4
NOTE: Once you become more confident in solving equations, you may be able to do two steps
at one time. However, in all of these examples, I will demonstrate all steps.
11) Solve 3[2m - (7 - 3m)] = m - 21.
3[2m - 7 + 3m] = m - 21
3[5m - 7] = m - 21
15m - 21 = m - 21
14m - 21 = -21
14m = 0
m = 0 is the solution.
simplify grouping symbols
subtract m on both sides
add 21 on both sides
divide both sides by 14
12) Solve 4 - 1x = 3x - 1 .
5 4
10
20 ∙ 4 - 20 ∙ 1x = 20 ∙ 3x - 20 ∙ 1 .
5
4
10
16 - 5x = 6x - 20
16 - 11x = -20
-11x = -36
x = 36/11 is the solution.
Multiply each term by LCD of 20.
subtract 6x on both sides
subtract 16 on both sides
divide both sides by -11
13) Solve
2
1 3
1
 3 x    5  x   3 x  7 
3
4 4
2
2
1
3
1
12   3 x    12  5  x   12  3x  7 
3
4
4
2
1

8 3 x    95  x   63 x  7 
4

24x - 2 = 45 - 9x - 18x + 42
24x - 2 = 87 - 27x
51x - 2 = 87
51x = 89
x = 89/51 is the solution.
14) Solve
Multiply each term by LCD of 12
simplify grouping symbols
collect like terms
add 27x on both sides
add 2 on both sides
divide both sides by 51
3 1 3
 
2x x 5
10x ∙ 3 – 10x ∙ 3 = 10x ∙ 2
2x
5
x
15 - 6x = 20
-6x = 5
x = -5/6 is the solution.
multiply each term by LCD of 10x
subtract 15 on both sides
divide both sides by -6
15) Solve 2x + 12 - 4 - 6x = 8 .
4
3
12 ∙ 2x + 12 - 12 ∙ 4 - 6x = 8 ∙ 12
4
3
3(2x + 12) - 4(4 - 6x) = 96
6x + 36 - 16 + 24x = 96
30x + 20 = 96
30x = 76
x = 76/30 is the solution.
multiply each term by LCD of 12
simplify grouping symbols
collect like terms
subtract 20 on both sides