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Section 1.2
Linear Equations
and
Rational Equations
Solving Linear Equations in
One Variable
Definition of a Linear Equation
A linear equation in one variable x is an equation
that can be written in the form
ax+b=0
where a and b are real numbers, and a  0.
Example
Solve and check:
7 x 10  18
Example
Solve and check:
2( x  5)  5(7  x)  9
Finding Intersection on a Graphing Calculator
2x  1   x  4
y1  2 x  1

y2   x  4
Let’s find the intersection of the
two lines. Press 2nd Trace to get
Calc. Then press #5 for
Intersection.
Questions
about curves
and quesses
Keep pressing enter
(three times) to
answer the questions
and the final result will
be the intersection
point (1,3).
Find the answer at the
bottom of the screen.
(1, 3)
Using your calculator and the intersection
function find the solution to the following problem.
2  3x 14  5x  2
Linear Equations with Fractions
Example
Solve and check:
x3 x7

2
2
6
Rational Equations
A rational equation is an equation
containing one or more rational
expressions. In the previous example we
saw a rational equation with constants in
the denominators. That rational equation
was a linear equation. The rational equation
below is not a linear equation. The solution
procedure still involves multiplying each
side by the least common denominator. We
must avoid any values of the variable that
make a denominator zero.
Example
Solve and check:
2x
6
4x

 2
x  4 x  4 x  16
Types of Equations
An equation that is true for all real numbers for
which both sides are defined is called an
identity. An example of an identity is
X+3=X+2+1
An equation that is not an identity, but that is
true for at least one real number, is called a
conditional equation.
2X=8
An inconsistent equation is an equation that is
not true for even one real number. An example
of an inconsistent equation is
X=X+7
Example
Solve and determine if the equation is an identity, a
conditional equation or an inconsistent equation.
3( x  5)  8  2 x
9  x  3  x  3  2 x
y  23  y  6
Solve and check:
5 x  7  8(4  x)
(a)
(b)
(c)
(d)
x2
x3
x  3
x  8
Solve the equation.
3
5
 4
x
x
(a)
(b)
(c)
(d)
x2
x3
x=5
x  3