Linear equations

LINEAR EQUATIONS
Prepared By:
Malik Sabah-ud-din
http://basic2advanced.blogspot.comWeb Site: 1

∗ Classify equations as linear, fractional, or rational,
∗ Solve linear equations,
∗ Solve equations leading to the form ax+b=0, and
∗ Solve application problems involving linear equations by
developing mathematical models for real-life problems.
GENERAL OBJECTIVE
At the end of the lesson the students are expected
to:

∗ Identify an equation,
∗ Classify equations as identity, conditional or equivalent,
∗ Distinguish a consistent from an inconsistent equation,
∗ Enumerate the properties of equality.
TODAY’S OBJECTIVE
to:

An equation is a statement that two mathematical expressions
are equivalent or equal.
DEFINITION
EQUATION
The values of the unknown that makes the equation true are
called solutions or roots of the equation, and the process of
finding the solution is called solving the equation.
Example:
9x2
=
117x =+ x32x37 −=−
5x32x7x4 +++=+
1
2x
x
2x
3x
+
+
=
−
+

∗ An identity equation is an equation that is true for any number
substituted to the variable.
KINDS OF EQUATIONS
121)(x.
3)3(.
3443.
22
2
++=+
−=−
+=+
xxc
xxxxb
xxa
Example:

∗ A conditional equation is an equation that is true only for
certain values of the unknown.
12)3(x.
0124.
232.
+=−
=−
−=+
xc
xb
xxa
Example:
Week 1 Day 1

∗ Two equations with exactly the same solutions are called
equivalent equations.
4.
2225.
205.
=
=+
=
xc
xb
xa
Example:
The following are equivalent equations.
Week 1 Day 1

∗ An inconsistent equation is an equation that has no solution.
∗ A consistent equation is an equation that has a solution.
Week 1 Day 1

EXAMPLE
Determine whether the given equation is an identity or a
conditional equation.
( )( )
( )
15
8
53
.5
1
3
1
3
1
.4
532
.3
96432x.2
339.1
22
2
xxx
x
x
xx
x
xxx
xx
xxx
=+
−
−
=
−
+
−
=+
++=+
+−=−
Week 1 Day 1

For all real numbers a , b and c
1. Addition Property of Equality
If a = b then a + c = b + c
2. Subtraction Property of Equality
If a = b then a – c = b – c
3. Multiplication Property of Equality
If a = b then a ∙ c = b ∙ c
4. Division Property of Equality
If a =b then
0cwhere
c
b
c
a
≠=
PROPERTIES OF EQUALITY
Week 1 Day 1

∗ Define linear equations in one variable,
∗ Determine the difference between linear and nonlinear
equations,
∗ Enumerate the steps in solving linear equations,
∗ Solve linear equations and equations involving fractions,
∗ Solve rational equations which are reducible to linear
equations,
∗ Define extraneous solution.
TODAY’S OBJECTIVE
to:
Week 1 Day 2

∗ An identity equation is an equation that is true for any number
substituted to the variable.
RECALL
• An equation is a statement that two mathematical
expressions are equivalent or equal.
• A conditional equation is an equation that is true only for
certain values of the unknown.
• Two equations with exactly the same solutions are called
equivalent equations.
• An inconsistent equation is an equation that has no solution.
• A consistent equation is an equation that has a solution.
Week 1 Day 2

DEFINITION
LINEAR EQUATION IN ONE VARIABLE
A linear equation in one variable is an equation that can
be written in the form
ax + b = 0
where a and b are real numbers and a  0
Example:
2x – 1 = 0, -5x = 10 + x, 3x + 8 = 2
Week 1 Day 2

Linear Equations Nonlinear Equations
354 =−x 822
=+ xx
7
2
1
2 −= xx 06 =− xx
3
6
x
x =− 12
3
=− x
x
Nonlinear; contains
the square of the
variable
Nonlinear; contains
the reciprocal of the
variable
Nonlinear; contains
the square root of
the variable
Week 1 Day 2

SOLVING A LINEAR EQUATION IN ONE VARIABLE
Steps
1. Simplify the algebraic expressions on both sides of the equation.
2. Gather all the variable terms on one side of the equation and
all constant terms on the other side.
3. Isolate the variable.
4. Check the solution by substituting the value of the
unknown into the original equation.
Week 1 Day 2

EXAMPLE
STEP DESCRIPTION EXAMPLE
1 Simplify the algebraic expression
on both sides
2(x-1)+3 = x-3(x+1)
2x-2+3 = x-3x-3
2x+1 = -2x-3
2 Gather all the variables on one
side of the equation and all
constant terms on the other
side.
2x+2x = -3-1
4x = -4
3 Isolate the variable
1-x
4
4
x
=
−
=
Problem #23 on page 97
Week 1 Day 2
Solve for the indicated variable: 2(x-1)+3=x-3(x+1)

Solve the following equations.
( )[ ] ( ) ( )[ ]3y31y55y232y3-5y2-25
97.pp
32#
+−−−−−=++
( )[ ] ( ) ( )[ ]y643y2627y472-6y98y-7-46
97.pp
36#
+−−−−−=+
Week 1 Day 2

Linear Equations Involving Fractions.
4
63
x2
7
x
97.pp
39#
+=
15
1x6
5
2x
3
5-x
-1
97.pp
48# −
−
+
=
Week 1 Day 2

SOLVING RATIONAL EQUATIONS THAT ARE
REDUCIBLE TO LINEAR EQUATIONS
A rational equation is an equation that contains one or more
rational expressions.
Extraneous solution are solutions that satisfy a transformed
equation but do not satisfy the original equation.
Steps
1. Determine any excluded values(denominator equals 0).
2. Multiply the equation by the LCD.
3. Solve the resulting linear equation.
4. Eliminate any extraneous solution.
Week 1 Day 2

a7
12
2
a
2
93.pp
1.1.4ex.Classroom
.1 =−
)4a(a
8
a
5
4-a
2
94.pp
1.1.5ex.Classroom
.2
−
=−
x3x
1
6x2
1
12-4x
1
95.pp
1.1.6ex.Classroom
.3 2
−
=
−
−
3x
1
5-2x
2
95.pp
1.1.7ex.Classroom
.4
+
−=
Solve the following equations.
4
2
1u
u
u
Edition2ndWatsonandRedlin,by Stewart
ryTrigonomet&Algebra
78page1.1exercise.5
=
+
−






EXAMPLE Week 1 Day 2

∗ Solve equations using radicals
∗ Solve absolute value
equations
∗ Solve literal equations
TODAY’S OBJECTIVE
to:
Week 1 Day 3

∗ Steps in solving linear equations
∗ A rational equation is an equation that contains one or more
rational expressions.
∗ Steps in solving rational equations.
∗ Extraneous solution are solutions that satisfy a transformed
equation but do not satisfy the original equation.
RECALL
Week 1 Day 3

SOLVING EQUATIONS USING RADICALS
solutionrealnohasequationtheaandevenisnif
aandevenisnifa
oddisnifax
solutionahasaxequationThe
n
n
n
,0
0x
<
≥±=
=
=
existnotdoesbecausesolutionrealnohasx
xsolutionrealoneonlyhasx
xsolutionsrealtwohasx
xsolutionrealoneonlyhasx
Examples
1616
232:32
216:16
232:32
:
44
55
44
55
−−=
−=−=−=
±=±==
===
Week 1 Day 3

123x2.1 =−
Solve each equations:
Week 1 Day 3
51x3.2 =+
126x25.3 =+−

ABSOLUTE VALUE EQUATIONS
DEFINITION The absolute value of a number a is given by
.linenumberrealtheonaandxbetweencetandistheisax
,generallyMore.originthetoaofcetandistherepresentsitthatand
0aifa
0aifa
a
−



<−
≥
=
Week 1 Day 3

Solve each equations: (examples on page 131)
1457x3.2
35x2.1
=+−
=−
Solve each equations:
2x31-x.17
1565x3.13
+=
=++
EXAMPLE
131pagefromExercise 





133pagefromExercise 





Week 1 Day 3

SOLVING FOR ONE VARIABLE IN TERMS OF THE OTHER
Many formulas in the sciences involve several variables,
and it is often necessary to express one of the variables in
terms of the others.
2
r
r
mM
GF
equationtheinMiablevatheforsolve
=
lhwhlwA
equationtheinwiablevatheforsolve
222
r
++=
72-71pagefromExample 





Week 1 Day 3

∗ LINEAR EQUATIONS ARE SOLVED BY :
SUMMARY
1. Simplifying the algebraic expressions on both sides of the equation.
2. Gathering all the variable terms on one side of the equation and
all constant terms on the other side.
3. Isolating the variable.
4. Checking the solution by substituting the value of the
unknown into the original equation.
• RATIONAL EQUATIONS ARE SOLVED BY :
1. Determining any excluded values(denominator equals 0).
2. Multiplying the equation by the LCD.
3. Solving the resulting linear equation.
4. Eliminating any extraneous solution.
Week 1 Day 3

CLASSWORK
HOMEWORK
#s 31,33,35,43,46,51,55,57,61, 65 page 97-98
#s 32, 34, 42, 60 page 97
Week 1 Day 3

APPLICATION INVOLVING LINEAR
EQUATIONS
Week 2 Day 1 Application Involving Linear Equations (Algebra and Trigonometry, Young 2nd
Edition, page 100-113).

∗ Develop mathematical models for real-life problems,
∗ Solve application problems involving common
formulas,
∗ Solve number problems,
∗ Solve digit problems,
∗ Solve geometric problems, and
∗ Solve money and coin problems.
TODAY’S OBJECTIVE
Week 2 Day 1

1. Read and analyze the problem carefully and make
sure you understand it.
2. Make a diagram or sketch, if possible.
3. Determine the unknown quantity. Choose a letter to
represent it.
4. Set up an equation. Assign a variable to represent
what you are asked to find.
5. Solve the equation for the unknown quantity.
6. Check the solution.
STEPS IN SOLVING WORD PROBLEMS
Week 2 Day 1

Start
Read and analyze
the problem
Make a diagram or
sketch if possible
Determine the
unknown quantity.
Did you set up
the equation?
Set up an equation,
assign variables to
represent what you
are asked to find.
A
no yes
A
Solve the equation
Check the solution
Is the unknown
solved?
no
yes
End
Week 2 Day 1

1. Find three consecutive odd integers so that the sum of
the
three integers is 5 less than 4 times the first.
(Example 2 page 102)
2. Find two consecutive even integer s so that 18 times the
smallest number is 2 more than 17 times the larger
number.
(Classroom Ex. 1.2.2 page 102)
NUMBER PROBLEMS
Week 2 Day 1

1. A rectangle 3 inches wide has the same area as a square with
9 inch sides. What are the dimensions of the rectangle?
(Your Turn problem page 103)
2. Consider two circles, a smaller one and a larger one. If the
larger has a radius that is 3 feet larger than that of the
smaller circle and the ratio of the circumferences is 2:1, what
are the radii of the two circles. (#21 page 110)
GEOMETRY PROBLEMS
Week 2 Day 1

1. In an integer between 10 and 100, the unit’s digit is 3 greater
than the ten’s digit. Find the integer, if it is 4 times as large
as the sum of its digits.
(from Internet Guide to Engineering Mathematics)
2. A certain two digit number is equal to 9 times the sum of its
digits. If 63 were subtracted from the number the digits
would be reversed. Find the number.
(from Internet Guide to Engineering Mathematics)
3. The sum of the digits of a two-digit number is 11. If we
interchange the digits then the new number formed is 45
less than the original. Find the original number.
(onlinemathlearning .com)
DIGIT PROBLEMS
Week 2 Day 1

1. A change purse contains an equal number of pennies, nickels
and dimes. The total value of the coins is $1.44. How many
of each type does the purse contain?
(# 25 page 89 Algebra and Trig. By Stewart, Redlin and Watson, 2nd
edition)
2. Mary has $3.00 in nickels, dimes and quarters. If she has
twice as many dimes as quarters and five more nickels than
dimes, how many coins of each type doe she have?
edition)
MONEY AND COIN PROBLEMS
Week 2 Day 1

∗ Solve investment problems,
∗ Solve age problems, and
∗ Solve mixture problems.
TODAY’S OBJECTIVE
to:
Week 2 Day 2

1. An ambitious 14-year old has saved $1,800 from chores and
odd jobs around the neighborhood. If he puts this money
into a CD that pays a simple interest rate of 4% a year, how
much money will he have in his CD at the end of 18 months?
2. Theresa earns a full athletic scholarship for college, and her
parents have given her the $20,000 they had saved to pay for
her college tuition. She decides to invest that money with an
overall goal of earning 11% interest. She wants to put some
the money in a low-risk investment that has been earning 8%
a year and the rest of the money in a medium-risk investment
that typically earns 12% a year. How much money should she
put in each investment to reach her goal?
(Example #5 page 105)
INVESTMENT PROBLEMS
Week 2 Day 2

1. A father is four times as old as his daughter. In 6 years, he
will be three times as old as she is now. How old is the
daughter now?
edition)
2. A movie star, unwilling to give his age, posed the following
riddle to a gossip columnist. “Seven years ago, I was eleven
times as old ad my daughter. Now I am four times as old as
she is.” How old is the star?
edition)
AGE PROBLEMS
Week 2 Day 2

1. A mechanic is working on the coolant system of a vehicle
with a capacity of 11.0 liters. Currently the system is filled
with coolant that is 45% ethylene glycol. How much fluid
must be drained and replaced with 100% ethylene glycol so
that the system will be filled with coolant that is 60%
ethylene glycol?
2. For a certain experiment, a student requires 100 ml of a
solution that is 8% HCl(hydrochloric acid). The storeroom
has only solutions that are 5% and 15% HCl. How many
milliliters of each available solution should be mixed to get a
100 ml of 8% HCl?
(# 33 page 111)
MIXTURE PROBLEMS
Week 2 Day 2

3. A cylinder contains 50 liters of a 60% chemical solution.
How much of this solution should be drained off and
replaced with a 40% solution to obtain a final strength of
46%?
(#30 page 37 Applied College Algebra and Trig. By Linda Davis 3rd
edition)
MIXTURE PROBLEMS
Week 2 Day 2

∗ Solve uniform motion problems,
∗ Solve work problems, and
∗ Solve clock problems.
TODAY’S OBJECTIVE
to:
Week 2 Day 3

1. You and your roommate decided to take a road trip to the
beach one weekend. You drove all the way to the beach at
an average speed of 60 mph. Your roommate drove all ath e
way back (on the same route, but with no traffic) at an
average rate of 75mph. If the total trip drive took a total of
9 hours, how many miles was the trip to the beach?
2. A Cessna 150 averages 150 mph in still air. With a tailwind it
is able to make a trip in 2 1/3 hours. Because of the
headwind, it is only able to make a return trip in 3 ½ hours.
What is the average wind speed?
(Your turn problem page 108)
UNIFORM MOTION PROBLEMS
Week 2 Day 3

3. A motorboat can maintain a constant speed of 16 mph
relative to the water. The boat makes a trip upstream to a
marina in 20 minutes. The return trip takes 15 minutes. What
is the speed of the current? (# 41 page 111)
4. On a trip Jerry drove a steady speed for 3 hours. An accident
slowed his speed by 30 mph for the last part of the trip. If
the 190-mile trip took 4 hours, what was his speed during
the first part of the trip?
edition)
UNIFORM MOTION PROBLEMS
Week 2 Day 3

1. Connie can clean her house in 2 hours. If Alvaro helps her,
they can clean the house in 1 hour and 15 minutes together.
How long would it take Alvaro to clean the house by
himself?
(Example #8 page 109)
2. Next-door neighbors Bob and Jim use hoses from both
houses to fill Bob’s swimming pool. They know it takes 18
hours using both hoses. They also knew that Bob’s hose,
used alone, takes 20% less time that Jim’s hose alone. How
much time is required to fill the pool by each hose alone?
(#48 page 91 Algebra and Trig. By Stewart, Redlin and Watson, 2nd
edition)
WORK PROBLEMS
Week 2 Day 3

3. It takes 7 people 12 hours to complete a job. If they worked
at the same rate, how many people would it take to
complete the job in 16 hours.
edition)
WORK PROBLEMS
Week 2 Day 3

1. What time after 8 o’ clock will the hands of the
continuously driven clock be opposite each other?
2. What time after 5:00 am will the hands of the continuously
driven clock extend in opposite direction?
3. What time after 3:00 pm will the hands of the
continuously driven clock are together for the first time?
4. What time after 4 o’ clock will the hands of the
continuously driven clock from a right angle?
CLOCK PROBLEMS
Week 2 Day 3

In real world many kinds of application problems can be
solved through modeling with linear equations. The following
procedure will help you develop the model. Some problems
require development of a mathematical model, while others
rely on common formulas.
1. Read and analyze the problem carefully and make sure you
understand it.
2. Make a diagram or sketch, if possible.
3. Determine the unknown quantity. Choose a letter to represent
it.
4.Set up an equation. Assign a variable to represent what you
are asked to find.
5.Solve the equation for the unknown quantity.
6.Check the solution.
SUMMARY
Week 2 Day 3

Linear equations

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