1.1 Linear Equations

1
Chapter 1 Section 1 Linear Equations

• In this section, we will
• Solve linear equations
• Solve equations that lead to linear equations
• Solve applied problems involving linear
  equations

2
A linear equation in one variable (say, x) is any
equation that can be written in the form ax b
c where a, b, c are real numbers and
linear equation expression
can be solved
cannot be solved (only simplified)
What does it mean to solve an equation?
1.1 Linear Equations Solve Linear Equations
3
Equations with the same solution set are called
equivalent equations. The following properties
will be used to isolate the variable on one side
of the equation.
If any quantity is added to (or subtracted
from) both sides of an equation, an equivalent
equation is formed.
Addition Prop. of Equality
Subtraction Prop. of Equality
Multiplication Prop. of Equality
Division Prop. of Equality
If both sides of an equation are multiplied (or
divided) by the same non-zero number, an
equivalent equation is formed.
1.1 Linear Equations Solve Linear Equations
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• Solving Linear Equations in One Variable
• If the equation contains fractions, multiply both
  sides by the magic number (LCM of the
  denominators) to clear fractions
• Use the Distributive Property to remove the
  parentheses (then combine the
  like-terms on each side of the equation)
• Use the Addition and Subtraction Properties to
  get all of the variables on one side of the
  equation together
  (and all of the numbers on the other side of the
  equation)
• Use the Multiplication and Division Properties to
  make the coefficient of the variable equal to 1
• Check your potential solution (by substituting
  your potential solution into the original
  equation to ensure that it satisfies it)
• If your potential solution does not
  check, you need to
• fix the error do not leave the
  incorrect answer!

Isolate the variable by undoing the operations
1.1 Linear Equations Solve Linear Equations
5
Examples Solve the equation and check your
result.
optional we will clear this equation of decimals
check
1.1 Linear Equations Solve Linear Equations
6
Examples Solve the equation and check your
result.
clear the parentheses first
check
1.1 Linear Equations Solve Linear Equations
7
Examples Solve the equation and check your
result.
we will clear this equation of fractions
check
1.1 Linear Equations Solve Linear Equations
8
Examples Solve the equation and check your
result.
we will clear this equation of fractions
check
1.1 Linear Equations Solve Linear Equations
9
Examples Solve the equation and check your
result.
we will clear this equation of fractions
check
1.1 Linear Equations Solve Linear Equations
10
Some equations lead to linear equations, once
they have been simplified. We will study two
types Type 1 Polynomial Equations that Lead to
Linear Equations
Examples Solve the equation and check your
result.
check
1.1 Linear Equations Solve Equations that Lead
to Linear Equations
11
Examples Solve the equation and check your
result.
check
1.1 Linear Equations Solve Equations that Lead
to Linear Equations
12

• Type 2 Rational Equations that Lead to Linear
  Equations
• we will first note any domain restrictions
• factor all denominators
• multiply both sides by the LCD of all of the
  rational expressions in the equation
• divide out any common factors
• solve the resulting linear equation
• check your potential solution

With a proportion, we can simply cross multiply.

What is a proportion?

• Warning Be sure to check your potential
  solution.
• A potential solution that fails to check can be a
  result of
• an incorrect solution that must be fixed or
• an extraneous solution that must be discarded

1.1 Linear Equations Solve Equations that Lead
to Linear Equations
13
Examples Solve the equation and check your
result.
check
note
1.1 Linear Equations Solve Equations that Lead
to Linear Equations
14
Examples Solve the equation and check your
result.
check
note
1.1 Linear Equations Solve Equations that Lead
to Linear Equations
15
Examples Solve the equation and check your
result.
check
note
1.1 Linear Equations Solve Equations that Lead
to Linear Equations
16
Examples Solve the equation and check your
result.
check
note
1.1 Linear Equations Solve Equations that Lead
to Linear Equations
17
Solving formulas for a specified variable
isolate the specified variable Examples Solve
each formula for the indicated variable.
1.1 Linear Equations Solve Applied Problems
Involving Linear Equations
18

• How to Solve a Word Problem
• Step 1 Read the problem until you understand it.
  
• What are we asked to find?
• What information is given?
• What vocabulary is being used?
• Step 2 Assign a variable to represent what you
  are looking for.
• Express any remaining unknown
  quantities in terms of this variable.
• Step 3 Make a list of all known facts and form
  an equation or inequality to solve.
• It may help to make a labeled
  diagram, table or chart, graph
• Step 4 Solve
• Step 5 State the solution in a complete sentence
  by mirroring the original question.
• Be sure to include units when
  necessary.
• Step 6 Check your result(s) in the words of the
  problem
• Does your solution make sense?

1.1 Linear Equations Solve Applied Problems
Involving Linear Equations
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Examples A total of 10,000 is to be divided
between Peter and Lois, with Lois to receive
3,000 less than Peter. How much will each
receive?
1.1 Linear Equations Solve Applied Problems
Involving Linear Equations
20
Examples Peter, who is paid time-and-a-half for
hours worked in excess of 40 hours, had a gross
weekly wages of 442 for 48 hours worked. What
is his regular hourly rate?
1.1 Linear Equations Solve Applied Problems
Involving Linear Equations
21
Examples Going into the final exam, which will
count as two tests, Meg has test scores of 80,
83, 71, 61, and 95. What score does Meg need on
the final in order to have an average score of 80
or higher?
1.1 Linear Equations Solve Applied Problems
Involving Linear Equations
22
To solve a word problem involving percents, we
must first change the percents to their decimal
equivalents. example Give the decimal equivalent
of 5.
Percent means per 100
To change a percent to its decimal equivalent,
move the decimal point two places to the left.
1.1 Linear Equations Solve Applied Problems
Involving Linear Equations
23
Examples A builder of new tract homes in Quahog,
Rhode Island, reduced the price of a model home
by 15. If the new price is 425,000, what was
its original price? How much is the price
reduction?
1.1 Linear Equations Solve Applied Problems
Involving Linear Equations
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• Independent Practice
• You learn math by doing math. The best way to
  learn math is to practice, practice, practice.
  The assigned homework examples provide you with
  an opportunity to practice. Be sure to complete
  every assigned problem (or more if you need
  additional practice). Check your answers to the
  odd-numbered problems in the back of the text to
  see whether you have correctly solved each
  problem rework all problems that are incorrect.
• Read pp. 86-93
• Homework pp. 94-86 23-49 odds, 53, 55,
• 57, 77-93
  odds

1.1 Linear Equations