Download 1.1 Linear Equations

1.1 Linear Equations

A linear equation
in one variable is
equivalent to an
equation of the form

To solve an
equation means to
find all the solutions
of the equation.
ax  b  0
where a and b are real numbers
and a  0.
Equivalent Equations

Two or more
 Examples of
equations that have
equivalent
precisely the same
equations:
solutions are called
3  x and x  3
equivalent
( x  2)  6  2 x  ( x  1) and
equations.
x  8  3x  1
3x  5  4 and (3x  5)  5  4  5
Solving a Linear Equation

Solve the equation:

3x  5  4
(3 x  5)  5  4  5
3x  9
3x 9

3 3
x3




We replace the original
equation by succession
of equivalent equations.
Add 5 to both sides.
Simplify.
Divide both sides by 3.
Simplify.
Solving a Linear Equation
Involving Fractions

Solve the equation:
x 1 x
 
5 2 6
x
 x 1
30     30 
6
5 2
x
1
x
30   30   30 
5
2
6
6 x  15  5 x
6 x  5 x  15  5 x  5 x
x  15  0
x  15  15  0  15
x  15





Multiply both sides by
the least common
denominator 30.
Be sure to multiply all
terms by 30.
Divide out common
factors.
Subtract 5x to get the
x-terms on the left.
Simplify.
Solve an Equation Involving
Rational Expressions

Solve:
4 9 7x  4
 
x 5
5x
4 9 7x  4
 
x 5
5x
4
 9 7x  4 
5x   5x  

x
5x 
5
4
9
7x  4
5x   5x   5x 
x
5
5x
20  9 x  (7 x  4)
20  9 x  7 x  4
20  2 x  4
20  4  2 x  4  4
16  2 x
16 2 x

2
2
8 x
This is the given
equation.
Multiply both sides by
5x (LCD).
Be sure to multiply all
terms by 5x.
Divide out the
common factors.
Simplify.
Subtract 4 from both
sides.
Divide both sides by
2.