Week 1 -2 Linear Equations (2)

MATH10 ALGEBRA

LINEAR EQUATIONS

Week 1 Day 1 Linear Equations (Algebra and Trigonometry, Young 2nd Edition, page 90-99)

GENERAL OBJECTIVE

• 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.

At the end of the lesson the students are expected to:

Week 1 Day 1

TODAY’S OBJECTIVE

• Identify an equation,• Classify equations as identity, conditional or equivalent,• Distinguish a consistent from an inconsistent equation,• Enumerate the properties of equality.


Week 1 Day 1

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:

9x 2

117x x32x37

5x32x7x4 1

2x

x

2x

3x

Week 1 Day 1

KINDS OF EQUATIONS

• An identity equation is an equation that is true for any number substituted to the variable.

121)(x .

3)3( .

3443 .

22

2

xxc

xxxxb

xxa

Example:

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• 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:

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• Two equations with exactly the same solutions are called equivalent equations.

4 .

2225 .

205 .

xc

xb

xa

Example: The following are equivalent equations.

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• An inconsistent equation is an equation that has no solution.

• A consistent equation is an equation that has a solution.

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EXAMPLEDetermine 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 .122

2

xxxx

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

0c where cb

ca

PROPERTIES OF EQUALITYWeek 1 Day 1

TODAY’S OBJECTIVE

• 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.


Week 1 Day 2

RECALL

• An identity equation is an equation that is true for any number substituted to the variable.

• 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

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Linear Equations Nonlinear Equations

354 x 822 xx

72

12 xx 06 xx

36

xx 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

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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- x4

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#

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Linear Equations Involving Fractions.

463

x2

7

x

97.pp

39#

15

1x6

5

2x

3

5-x-1

97.pp

48#

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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.

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a7

122

a

2

93.pp

1.1.4 ex. Classroom .1

)4a(a

8

a

5

4-a

2

94.pp


x3x

1

6x2

1

12-4x

1

95.pp


2

3x

1

5-2x

2

95.pp


Solve the following equations.

4

2

1uu

u

Edition 2nd Watson and Redlin,by Stewart

ry Trigonomet & Algebra 78 page 1.1 exercise .5

EXAMPLE Week 1 Day 2

TODAY’S OBJECTIVE

• Solve equations using radicals• Solve absolute value equations• Solve literal equations


Week 1 Day 3

RECALL

• 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.

Week 1 Day 3

SOLVING EQUATIONS USING RADICALS

solutionrealnohasequationtheaandevenisnif

aandevenisnifa

oddisnifax

solutionahasaxequationThe

n

n

n

,0

0 x

existnotdoesbecausesolutionrealnohasx

xsolutionrealoneonlyhasx

xsolutionsrealtwohasx

xsolutionrealoneonlyhasx

Examples

16 16

232: 32

216: 16

232 : 32

:

44

55

44

55

Week 1 Day 3

123x2 .1

Solve each equations:

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51x3.2

126x25 .3

ABSOLUTE VALUE EQUATIONS

DEFINITION The absolute value of a number a is given by

.line number real the on a and x between cetandis the is ax

,generally More .origin the to a of cetandis the represents it that and

0a if a

0a if a 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


ry Trigonomet & Algebra 131 page from Exercise


ry Trigonomet & Algebra133 page from Exercise

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

mMGF

equationtheinMiablevatheforsolve

lhwhlwA

equationtheinwiablevatheforsolve

222

r


ry Trigonomet & Algebra 72-71 page from Example

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SUMMARY • LINEAR EQUATIONS ARE SOLVED BY :

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).

TODAY’S OBJECTIVE

• 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.

Week 2 Day 1

STEPS IN SOLVING WORD PROBLEMS

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.

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.

Ano yes

A

Solve the equation

Check the solution

Is the unknown solved?

no

yes

End

Week 2 Day 1

NUMBER PROBLEMS

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)

Week 2 Day 1

GEOMETRY PROBLEMS

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)

Week 2 Day 1

DIGIT PROBLEMS

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)

Week 2 Day 1

MONEY AND COIN PROBLEMS

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?


Week 2 Day 1

TODAY’S OBJECTIVE

• Solve investment problems,• Solve age problems, and • Solve mixture problems.


Week 2 Day 2

INVESTMENT PROBLEMS

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)

Week 2 Day 2

AGE PROBLEMS

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?


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?


Week 2 Day 2

MIXTURE PROBLEMS

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)

Week 2 Day 2

MIXTURE PROBLEMS

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)

Week 2 Day 2

TODAY’S OBJECTIVE

• Solve uniform motion problems,• Solve work problems, and• Solve clock problems.


Week 2 Day 3

UNIFORM MOTION PROBLEMS

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?

(Classroom Ex. 1.2.7 page 108) 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)

Week 2 Day 3

UNIFORM MOTION PROBLEMS

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?


Week 2 Day 3

WORK PROBLEMS

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)

Week 2 Day 3

WORK PROBLEMS

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.


Week 2 Day 3

CLOCK PROBLEMS

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?

Week 2 Day 3

SUMMARY

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.

Week 2 Day 3

CLASSWORK

HOMEWORK

#s 15,19,31,34,38,42,44,47,50,73 page 110-113

Classroom example 1.2.6 page 106 and 1.2.7 page 108

Week 2 Day 3

Week 1 -2 Linear Equations (2)

Documents

x 5x3 x2 x

equation true

identity equation

conditional equation

inconsistent equation

original equation

definitionlinear equation

definitionequationan