Solving Linear Equations
Introduction
An equation is a statement that two mathematical expressions
are equal. 2x + 4 = 8 is an equation.
To solve an equation involving x means you are trying to find
all the values of x that make the equation true.
The solution set is the set of all real numbers that are
solutions to the equation.
If every real number solves an equation, the equation is
called an identity. For example, 2(x 1) = 2x 
2 is an identity equation because any real number you substitute
for x makes this equation true.
If only specific real numbers solve an equation, the equation
is called a conditional equation. For example,
x + 4 = 6 is true if and only if x = 2 so it is a conditional
equation.
An equation that has no solution is said to have an empty set
solution. For example, + 1 = 0 has no real number solutions. We will use to indicate the empty set.
You check a solution set by merely substituting your answer in
the original equation to verify your results.
Example: Solve x + 4 = 6
Answer x = 2
To check: Substitute 2 everywhere you have an x.
And the solution checks out fine!
Trial Solutions
See if 5 satisfies the equation 2x + 3 = 10
To check 5 in the equation, substitute it wherever you have an
x.
2x + 3 = 10 Original equation
Substitute 5 for x
So 5 is not a solution to this equation!!!!
Two equations that have the same solution set are called equivalent
equations.
To solve linear equations, you are trying to work your way to
the simple equation
x = ______
 Remove any parentheses in the equation.
 Then try to get all the unknown terms (those involving x
) on one side of the equation and all the constant values
on the other side. I always put the x term on the left
hand side of the equation and the constant terms on the
right hand side of the equation. Remember  you can do
"almost" anything to an equation as long as you
do it to both sides of the equation!! That means you can
add the same quantity to both sides; you can subtract the
same quantity from both sides; you can multiply both
sides of an equation through by the same number; and you
can divide both sides of an equation through by the same
number.
 Collect like terms
 And remember, it is okay to completely swap each side of
the equation. 2 = x is the same as x = 2 !!!
A linear equation in one variable x is an
equation that can be written in the standard form ax + b = c. It
is called a firstdegree equation because its variable has an
exponent of 1. A linear equation should always have exactly one
solution.
Examples
Solve the following standard linear equation for x and check
your answer by substitution.
3x  7 = 2 

3x 7 + 7 = 2 + 7 
Add 7 to both sides 
3x = 9 
Collect like terms 

Now divide both sides by 3 
x = 3 
This is your answer 
CHECK: Substitute 3 for each x in the original equation
and the solution checks !!!
