Download Linear Equations in One Variable

Linear Equations in One Variable
MATH 101 College Algebra
J. Robert Buchanan
Department of Mathematics
Summer 2012
J. Robert Buchanan
Linear Equations in One Variable
Objectives
In this section we will learn how to:
Recognize and combine like terms.
Solve linear equations of the form: ax + b = c.
Solve absolute value equations of the form: |ax + b| = c.
J. Robert Buchanan
Linear Equations in One Variable
Recognizing Like Terms
A term in an equation is an expression combining variables
and constants using the operations of multiplication and
division.
A term containing only a real number is called a constant term
or just a constant.
The numerical factor of a non-constant term is called a
numerical coefficient or just the coefficient.
If no explicit coefficient is written, it is assumed to be 1.
If a negative sign precedes a non-constant term without an
explicit numerical coefficient, the coefficient is assumed to
be −1.
If an equation contains more than one term, like terms have
the same variables raised to the same powers.
J. Robert Buchanan
Linear Equations in One Variable
Combining Like Terms
To combine like terms, add or subtract the coefficients and
keep the common variable expression.
Unlike terms cannot be combined.
J. Robert Buchanan
Linear Equations in One Variable
Linear Equations: ax + b = c
Terminology:
An algebraic expression is a combination of variables
and numbers using the operations of addition, subtraction,
multiplication, and division.
An equation is a statement that two algebraic expressions
are equal.
If an equation contains a variable, any number which
makes the equation true when substituted for the variable
is called a solution.
The set of all numbers making an equation true is called a
solution set.
Equations with the same solution sets are said to be
equivalent.
An equation of the form
ax + b = c
with a 6= 0 is called a linear equation in x.
J. Robert Buchanan
Linear Equations in One Variable
Solving Linear Equations (1 of 2)
Equations are solved by using the following properties of
equations.
Addition/Subtraction Property: if the same expression is added
to (or subtracted from) both sides of an equation,
the two equations are equivalent.
A = B
A+C = B+C
A−C = B−C
Multiplication/Division Property: if both sides of an equation are
multiplied (or divided) by the same non-zero
expression, the two equations are equivalent.
A = B
AC = BC
B
A
=
C
C
if C 6= J.0.Robert Buchanan
Linear Equations in One Variable
Solving Linear Equations (2 of 2)
To solve a linear equation:
1
Simplify each side of the equation by removing grouping
symbols and combining like terms.
2
Use the Addition/Subtraction Property to add the opposites
of constants or variable expressions so that the variable
expressions are on one side of the equation and the
constants are on the other.
3
Use the Multiplication/Division Property to multiply both
sides of the equation by the reciprocal of the coefficient of
the variable, so that the new coefficient is 1.
4
Check your answer by substituting it into the original
equation.
J. Robert Buchanan
Linear Equations in One Variable
Types of Equations
Equations can be classified depending on the number of
solution they possess.
If an equation has a finite number of solutions, it is called a
conditional equation.
If an equation is solved by every real number in R then the
equation is called an identity.
If an equation has no solutions (the solution set is the
empty set ∅) the equation is called a contradiction.
J. Robert Buchanan
Linear Equations in One Variable
Absolute Value
Definition
For any real number x,
|x| =
x
−x
if x ≥ 0,
if x < 0.
Solving equations involving absolute value.
If |x| = c > 0 then x = c or x = −c.
If|ax + b| = c > 0 then ax + b = c or ax + b = −c.
If |a| = |b| then a = b or a = −b.
If|ax + b| = |cx + d| then ax + b = cx + d or
ax + b = −(cx + d).
J. Robert Buchanan
Linear Equations in One Variable