Math 101 Intermediate Algebra

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 ab = cd 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 numbers--has 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 first-degree 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
 x = ba

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
 x = ba
Check your solution in the original equation by substitution.