Download Solving Linear Equations

The LHS and RHS are each
expressions which are simplified
according to PEMDAS.
Initial Simplification Stage
Math 60
Solving Linear Equations
Elementary Algebra
General Four–Step LINEAR* Equation Solving Process**
Simplify the Left Hand Side (LHS):
 Distribute first, if needed.
 Then, Combine Like Terms (CLT), if needed.
 When done you should have no more than 1 Linear* Variable Term and 1 Constant Term on
the LHS.
Simplify the Right Hand Side (RHS):
 Distribute first, if needed.
 Then, Combine Like Terms (CLT), if needed.
 When done you should have no more than 1 Linear* Variable Term and 1 Constant Term on
the RHS.
Solution Stage
Although it is not required,
it helps to move the variable
terms to the left and the
constant terms to the right.
ONCE EACH SIDE IS FULLY SIMPLIFIED, WE ARE READY TO SOLVE BY
WORKING WITH BOTH SIDES OF THE EQUATION SIMOTANEOUSLY.
Undo Add/Subtract first.
 Addition is undone with Subtraction.
 Subtraction is undone with Addition.
Undo Multiply/Divide Last.
 Multiplication is undone with Division.
 Division is undone with Multiplication.
* The word “linear’ refers to an exponent of 1 on the variable base, in a base–exponent pair. That is, the equation contains an x1 = x, or
a y1 = y etc. The letter used to represent the variable does not matter. What matters is the exponent of 1.
**Occasionally we may need to start by multiplying both sides of the equation by the LCD to clear all the fractions. Then it will be
possible to do simplify both the left and right hand sides in an easier manner without using fractions.
Comments about the format of your answer:
 The type of answer you return (whole number, integer, fraction, or decimal) and they way you carry out your
computations (by hand or on calculator), will be consistent with the type of numbers present in the original equation.
 Equations containing whole number, integers, or fractions will return as a whole number, integer, or fraction.
o Whole numbers and integers are fractions, so basically an equation with fractional coefficients and fractional
constant terms returns a fractional answer. All fractional answers must be in lowest terms.
o For whole number, integers, and fractions the work is done by hand without the use of a calculator.
 Equations contacting decimals return decimal answers, and the work may be done with the aid of a scientific calculator.
Decimal answers may sometimes need to be rounded.
Practice problems: Solve. You will need to do your work on another sheet of paper.
1.
10  x  6
2.
2 x  4 x  1
3.
x x
 2
5 3
4.
5
 x  4  6
9
5.
4  y  6   1  11
6.
3  4  x  2   5  3x
7.
6  3x  5 1  x   3  x  2   4
8.
2  x  7   5x  12  5  2  x 
9.
2x  3  9 x  3x  3 11x
10.

11.
x x
x
   2
4 3
5
12.
0.37 x  2.5 11  x   5  x  0.26
2
5
 2z  9 1   z 
3
9
End of Review Material
(For #12: Round your answer to the nearest tenth.)
Start of New Material
Most of the time when we solve a linear equation we produce exactly one solution. An equation with exactly one solution
is called a Conditional Equation. However, sometimes when we solve equations all the variables drop out and we are left
with either a True Statement or a False Statement. When this happens we produce one of two types of special solutions.
True Statement 
This is also called an Identity.
All values of x satisfy this relationship.
Answer can be written as:
This is also called a Contradiction.
No values of x satisfy this relationship.
Answer can be written as:
, All Real Numbers,  ,   ,or
x |
False Statement  No Solution
(All Real Numbers)
No Solution, {}, or 
x is a real number
Example:
3  x  1  9  8x  6  5x
10  x  10  5  5  2 x  20 
Example:
There are only three possible
answers you can get when
solving a linear equation:
1. x = ?
2. All Real Numbers
3. No Solution
3x  6 = 3x  6
6 =6
(T)
All Real Numbers
10 x  95 = 10 x  100
95 = 100
(F)
No Solution
CAUTION: You must continue to solve the equation by showing that the variable terms drop out and a statement (either
True or False) consisting of only constant terms is obtained.
Practice problems: Solve. You will need to do your work on another sheet of paper.
13.
2  x  4   6  2 x
14.
8x  3  2x  6x  3  4 x
16.
9  y  1  3 y  2  3 y  1  8
17.
4
3
19.
5  x  2  1   2  x  5  9
20.
Answers:
1.
x4
15.
4   x  3  2  3x  5  4  10 x
 2 x 1  3x  x  32 3x  2
18.
3  x  2.1  9.7  16.6  x  4  5  x 
3x  5   2 x  1    x  6 
21.
4  4  5x  1  4  3x  1  4 1  2 x 
2.
x   16
3.
x  15
4.
x  185
5.
y  3
6.
x6
7.
z
11.
x  120
23
12.
x  10.3
13.

14.
19.

20.
8.
x  8
9.
x0
10.
15.

16.

17.
40
7
18.
21.
x  15