Solving Equations
Chapter 2, Section 1
 Properties of Equality
 For all real numbers a, b, c:

Reflexive Property

a = a

Symmetric Property

If a = b, then b = a

Transitive Property

If a = b and b = c, then a = c

Addition Property of Equality

If a = b, then a + c = b + c

Multiplication Property of Equality

If a = b, then a x c = b x c

Proportions
(cross multiply)

If

a b

=

c d

with b, d 0,

then ad = bc.


 Combining Terms
 Defn: The coefficient is the numerical part of a term that
precedes the variable.
 Defn: The degree of a term is the sum of the exponents on the variables
in the term.
 Defn: Like terms have the same variables with the same
exponents.
 To simplify an expression means to combine all like terms in the expression.
 Equations
 Defn: A solution (or root) of an equation is a number that
makes the equation
true when that number is substituted in for the variable.
 Equations may have one solution, no solution, or many solutions:
Conditional Equation

Has exactly one real solution.

Identity

Is true for all real numbershas an infinite
number of solutions.

Inconsistent Equation

Has no solution

 Solving Linear Equations in One Variable
 Defn: A linear equation in one variable is a firstdegree equation
(largest exponent on the variable is 1) with only one variable.
 A linear equation in one variable may always be written in the form ax = b.
 Trick to solving: Use the properties of equality to
get the given equation into an equivalent equation of the
form ax = b. Then the solution is
Steps to Solving a Linear Equation

Eliminate fractions by multiplying both sides by the
least common denominator.

Remove grouping symbols
(as in "order of operations," Chapter1, section4).

Combine like terms on each side of the equal sign.

Use addition property of equality (maybe repeatedly) to
get the equation into the form ax = b.

Divide both sides by a.
The solution is

Check your solution in the original
equation by substitution.

