This document discusses equivalence and solving equations. It begins with defining equivalence as the product of a number and its multiplicative inverse being 1. It then provides examples of solving various equations using properties of equality like addition, subtraction, multiplication and division. These include one-step, two-step and multi-step equations with or without variables on both sides. It also discusses equations having no solutions, one solution, or infinitely many solutions.
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The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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2. To
• solve equations using the Inverse
Property of Multiplication
• solve equations with rational coefficients
Course 3, Lesson 2-1
Expressions and Equations
3. Course 3, Lesson 2-1
Expressions and Equations
Words The product of a number and its multiplicative inverse is 1.
Numbers
Symbols
7 8 1
78
3 2 1
2 3
1, where and 0
a b
a b
b a
4. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
1. Solve c = 18. Check your solution.
7
8
c = 18
c = 18
c = 24
Check c = 18
(24) = 18
18 = 18
Write the equation.
Multiply each side by the multiplicative inverse of , .
Write 18 as . Divide by common factors.
Simplify.
Write the original equation.
Replace c with 24.
Write 24 as . Divide by common factors.
611
1 1 1
?
6
1
This sentence is true.
6. 1
Need Another Example?
2
3
4
5
Step-by-Step Example
2. Solve 1 s = 16 . Check your solution.
s = 11
Write the equation.
Simplify.
Multiply each side by the multiplicative inverse of , .
Divide by common factors.
11 111
11 11
8. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
3. Solve 3.15 = 0.45n. Check your solution.
3.15 = 0.45(7)
Write the equation.
Replace n with 7.
Division Property of Equality
Simplify.
3.15 = 0.45n
7 = n
Check 3.15 = 0.45n Write the original equation.
3.15 = 3.15 This sentence is true.
10. 1
Need Another Example?
2
3
4
5
Step-by-Step Example
4. Latoya’s softball team won 75%, or 18, of its games.
Define a variable. Then write and solve an equation to
determine the number of games the team played.
n = 24 Simplify.
Write the equation. Write 75% as 0.75.
Division Property of Equality
Latoya’s softball team won 18 games, which was 75% of the
games played. Let n represent the number of games played.
Write and solve an equation.
0.75n = 18
Latoya’s softball team played 24 games.
11. Answer
Need Another Example?
Antonio has some fabric that he will use to make
curtains. Forty-five percent, or 6 yards, of the
fabric is green. Define a variable. Then write and
solve an equation to determine how many yards
of fabric he has altogether.
0.45n = 6; 13 yd
12. To
• identify the Properties of Equality
• solve two-step equations
Course 3, Lesson 2-2
Expressions and Equations
14. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
1. Solve 2x + 3 = 7.
There are two 1-tiles in each group, so x = 2.
Write the equation.
Subtraction Property of Equality
Division Property of Equality
Remove three 1-tiles from each mat.
Separate the remaining tiles into 2 equal groups.
Using either method, the solution is 2.
7
Use a model.
2x + 3 – 3 = 7 – 3
2x = 4
Use symbols.
2x + 3 = 7
Simplify.x = 2.
–3 = –3
2x = 4
16. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
2. Solve 25 = n – 3.
Write the equation.
Addition Property of Equality
Multiplication Property of Equality
25 = n – 3
28 = n
The solution is 112.
112 = n
Simplify.
+3 = +3
20. 1
Need Another Example?
2
3
4
5
Step-by-Step Example
4. Chicago’s lowest recorded temperature in degrees
Fahrenheit is –27°. Solve the equation –27 = 1.8C + 32 to
convert to degrees Celsius.
Write the equation.
Division Property of Equality
–32.8 ≈ C
Simplify.
Subtraction Property of Equality
–27 = 1.8C + 32
Simplify. Check the solution.
So, Chicago’s lowest recorded temperature is about
–32.8 degrees Celsius.
–32 = –32
–59 = 1.8C
21. Answer
Need Another Example?
Melisa wants to put trim molding around a
rectangular table. The table is 45 inches long and
she has 150 inches of trim. Solve the equation
150 = 2w + 90 to find the width of the table.
30 in.
22. To
• translate sentences into equations
• define a variable, then write and
solve equations
Course 3, Lesson 2-3
Expressions and Equations
23. 1
Need Another Example?
2
Step-by-Step Example
1. Translate the sentence into an equation.
Eight less than three times a number is –23.
Eight less than three times a number is –23.Words
3
Let n represent the number.Variable
3n – 8 = –23Equation
25. 1
Need Another Example?
2
3
Step-by-Step Example
2. Translate the sentence into an equation.
Thirteen is 7 more than one-fifth of a number.
Thirteen is 7 more than one-fifth of a number.Words
Let n represent the number.Variable
Equation 13 = n + 7
27. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
3. You buy 3 books that each cost the same amount and
a magazine, all for $55.99. You know that the magazine
costs $1.99. How much does each book cost?
3b + 1.99 = 55.99
So, the books each cost $18.7
Three books and a magazine cost $55.99.Words
Let b represent the cost of one book.Variable
3b + 1.99 = 55.99Equation
Write the equation.
Subtraction Property of Equality
Simplify.3b = 54.00
Division Property of Equality
b = 18 Simplify.
–1.99 = – 1.99
28. Answer
Need Another Example?
Kendra paid $7 for her admission ticket to the fair and
bought 12 ride tickets. She spent a total of $31 on
admission and ride tickets. Write and solve an equation to
find the cost of one ride ticket.
7 + 12x = 31; $2
29. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
4. A personal trainer buys a weight bench for $500 and w
weights for $24.99 each. The total cost of the purchase
is $849.86. How many weights were purchased?
500 + 24.99w = 849.86
Bench plus $24.99 per weight equals $849.86Words
Let w represent the number of weights.Variable
500 + 24.99 • w = 849.86Equation
Write the equation.
Subtraction Property of Equality
Simplify.24.99w = 349.86
Division Property of Equality
w = 14 Simplify.
So, 14 weights were purchased.
–500 = – 500
30. Answer
Need Another Example?
Membership at the health club is $15 per month
plus $10 for each exercise class taken. Jill paid
$75 for the month of September. Write and solve
an equation to find the number of exercise
classes Jill took in September.
15 + 10c = 75; 6 classes
31. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
5. Your and your friend’s lunch cost $19. Your lunch
cost $3 more than your friend’s. How much was
your friend’s lunch?
f + f + 3 = 19
Your friend’s lunch plus your lunch equals $19.Words
Let f represent the cost of your friend’s lunch.Variable
f + f + 3 = 19Equation
Write the equation.
Subtraction Property of Equality
Simplify.2f = 16
Division Property of Equality
f = 8 Simplify.
Your friend spent $8.
2f + 3 = 19 f + f = 2f
–3 = – 3
32. Answer
Need Another Example?
You and your friend spent a total of $33 for dinner. Your
dinner cost $5 less than your friend’s. Write and solve an
equation to find how much you spent for dinner.
d + (d – 5) = 33; $14
33. To solve equations with
• variables on both sides
• rational coefficients
Course 3, Lesson 2-4
Expressions and Equations
34. 1
Need Another Example?
2
3
4
Step-by-Step Example
1. Solve 8 + 4d = 5d. Check your solution.
8 + 4d = 5d Write the equation.
Write the original equation.Check 8 + 4d = 5d
8 = d
Subtraction Property of Equality
To check your solution, replace d with 8 in the original equation.
Simplify by combining like terms.
Subtract 4d from the left side of
the equation to isolate the variable.
Subtract 4d from the right side of
the equation to keep it balanced.
Replace d with 8.
The sentence is true.
8 + 4(8) = 5(8)
40 = 40
?
–4d = – 4d
38. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
3. Green’s Gym charges a one time fee of $50 plus $30 per session for a personal
trainer. A new fitness center charges a yearly fee of $250 plus $10 for each session
with a trainer. For how many sessions is the cost of the two plans the same?
20s = 200
Simplify.
Check Green’s Gym: $50 plus 10 sessions at $30 per session
Simplify.
Addition Property of Equality
50 + 10 • 30 = 50 + 300
250 + 10 • 10 = 250 + 100
50 + 20s = 250
Subtraction Property of Equality
Write the equation.50 + 30s = 250 + 10s
fee of $50 plus
$30 per session
Words
Let s represent the number of sessions.Variable
50 + 30s = 250 + 10sEquation
is the
same as
a fee of $250 plus
$10 per session
Division Property of Equality
s = 10
So, the cost is the same for 10 personal trainer sessions.
Simplify.
= $350
new fitness center: $250 plus 10 sessions at $10 per session
= $350
– 10s = – 10s
– 50 = – 50
39. Answer
Need Another Example?
The measure of an angle is 8 degrees more than its
complement. If x represents the measure of the angle and
90 – x represents the measure of its complement, what is
the measure of the angle?
49°
40. 1
Need Another Example?
2
3
4
Step-by-Step Example
4. Solve x – 1 = 9 – x.
x = 12
Simplify.
Multiplicative Inverse Property
Addition Property of Equality
The common denominator of the coefficients is 6.
Rewrite the equation.
Addition Property of Equality
Simplify.
Simplify.
+ 1 = + 1
42. To solve
• multi-step equations,
• equations with no solutions,
• equations with an infinite
number of solutions
Course 3, Lesson 2-5
Expressions and Equations
43. Symbol
• null set Ø
• empty set { }
• identity
Course 3, Lesson 2-5
Expressions and Equations
44. Course 3, Lesson 2-5
Expressions and Equations
Null Set One Solution Identify
Words no solution one solution infinitely many solutions
Symbols a = b x = a a = a
Example 3x + 4 = 3x 2x = 20 4x + 2 = 4x + 2
4 = 0 x = 10 2 = 2
Since 4 ≠ 0, Since 2 = 2, the
there is no solution is all
solution. numbers.
45. Need Another Example?
Step-by-Step Example
1. Solve 15(20 + d) = 420.
1
2
3
4
15(20 + d) = 420 Write the equation.
Subtraction Property of Equality
Simplify.
300 + 15d = 420 Distributive Property
Simplify.15d = 120
Division Property of Equality
d = 8
– 300 = – 300
51. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
4. At the fair, Hunter bought 3 snacks and 10 ride tickets.
Each ride ticket costs $1.50 less than a snack. If he spent
a total of $24.00, what was the cost of each snack?
Write an equation to represent the problem.
Write the equation.
Simplify.
3s + 10(s – 1.5) = 24
Addition Property of Equality
13s = 39
3s + 10s – 15 = 24 Distributive Property
13s – 15 = 24 Collect like terms.
Division Property of Equality
Simplify.s = 3
So, the cost of each snack was $3.
+ 15 = + 15
52. Answer
Need Another Example?
The length of Philip’s stride when walking is 4 inches
greater than the length of Anne’s stride. If it takes Philip
5 steps and Anne 6 steps to walk the same distance,
what is the length of Anne’s stride?
20 in.
