Download Sullivan College Algebra: Section 1.1

Sullivan Algebra and
Trigonometry: Section 1.1
Linear Equations
Objectives of this Section
• Solve a Linear Equation
• Solve Equations That Lead to Linear Equations
• Solve Applied Problems Involving Linear Equations
The set of all values of a variable that make
an equation a true statement are called
solutions, or roots, of the equation.
Example: The statement x + 5 = 9 is true
when x = 4 and false for any other choice
of x. Thus, 4 is a solution of the equation.
We also say that 4 satisfies the equation x
+ 5 = 9.
We solve equations by creating a series of
equivalent equations, that is equations that
have precisely the same solution set.
Example: The equations 3x – 4 = 5x
and 2x = -4 are equivalent equations,
since the solution set for both equation
is {-2}.
Procedures that Result in Equivalent
Equations
1. Interchange the two sides of the equation.
2. Simplify the sides of the equation by combining
like terms, eliminating parenthesis, and so on.
3. Add or subtract the same expression on both sides
of the equation.
4. Multiply or divide both sides of the equation by
the same nonzero expression.
5. If one side of the equation is 0 and the other side
can be factored, then set each factor equal to 0.
Example


Solve: 3x x  3  3 x  3

2

3 x  x  3  3 x  3
2
3x  9 x  3x  9
2
2
 9 x  9
x 1
Example
2x
8
4
Solve:
x 1
x 1
2x
8
 4 x  1 
 x  1
 x  1
x 1
x 1
2x  4x  4  8
6x  4  8
6 x  12
x2
Example
Solve the equation:
x 3x 1 x  2x  4
x  2x  3  x  6x  8
 2 x  3  6x  8
 2 x  6x  11
 8 x  11
2
2
11
x  
8
A linear equation in one variable is equivalent
to an equation of the form:
ax + b = 0
Note that the previous example,
x 3x 1x 2x 4
Is an example of a linear equation, since it
is equivalent to:
8 x + 11 = 0
Both equations have the same solution set: { - 118 }
Example
Solve the equation:
2(x - 3) = 5x - 9
2x - 6 = 5x - 9
2x = 5x - 9 + 6
-3x = - 3
x=1
Steps for Setting Up Applied Problems
Step 1: Read the problem carefully, perhaps two or
three times. Identify what you are looking for.
Step 2: Assign a letter (variable) to represent what
you are looking for. Express any remaining
unknown quantities in terms of that variable.
Step 3: Make a list of known facts and translate
them into mathematical expressions. These may
take the form of an equation or later, an inequality.
Step 4: Solve the equation for the variable and
answer the question using a sentence.
Step 5: Check your answer with the facts of the
problem.
Example: Yolanda, Mary, and Sophie won $200,000
playing the lottery. Based on how much each
contributed to buy the ticket, Mary gets four fifths of
what Yolanda gets, while Sophie gets three fourth of
what Mary gets. How much does each receive?
y: amount received by Yolanda
m: amount received by Mary
s: amount received by Sophie
y  m  s  200000
y  m  s  200000
4
m y
5
3
s m
4
3 4 
  y
45 
3
 y
5
4
3
y  y  y  200000
5
5
12
y  200000
5
y  $83,333.33
4
m  ($83,333.33)  $66,666.67
5
3
s  ($83,333.33)  $50,000
5