CBSE Class 10 Mathematics Real Numbers Topic
Real Numbers Topics discussed in this document:
Introduction
Rational numbers
Fundamental theorem of Arithmetic
Decimal representation of Rational numbers
Terminating decimal
Non-terminating repeating decimals
Irrational numbers
Surd
General form of a surd
Operations on surds
· Addition and subtraction
· Multiplication of surds
More Topics under Class 10th Real Numbers (CBSE):
Real numbers
Laws of
logarithms
Common and natural logarithms
Visit Edvie.com for more topics
This presentation is intended to help the education students by giving an idea on how they will use technology in education specially in teaching mathematics.
CBSE Class 10 Mathematics Real Numbers Topic
Real Numbers Topics discussed in this document:
Introduction
Rational numbers
Fundamental theorem of Arithmetic
Decimal representation of Rational numbers
Terminating decimal
Non-terminating repeating decimals
Irrational numbers
Surd
General form of a surd
Operations on surds
· Addition and subtraction
· Multiplication of surds
More Topics under Class 10th Real Numbers (CBSE):
Real numbers
Laws of
logarithms
Common and natural logarithms
Visit Edvie.com for more topics
This presentation is intended to help the education students by giving an idea on how they will use technology in education specially in teaching mathematics.
debatavond ‘A little less conversation’,
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- VVE, de Vereniging voor Economie
- Rotary Club Oostrozebeke-Mandeldal
panelleden:
Tom De Volder (advocaat), Stefaan Lammertyn (online marketeer), Jan De Cock (journalist)
moderator:
Veerle De Jaegher (social media marketeer)
This is a PPT created and developed by Alankrit Wadhwa of Army Public School, Pune. He made this PPT with great effort and is credible for the same. I hope this PPT makes this chapter a lot more fun and easier to understand.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
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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.
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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.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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This will be used as part of your Personal Professional Portfolio once graded.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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Unit 8 - Information and Communication Technology (Paper I).pdf
Unit6: Algebraic expressions and equations
1. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Algebraic expressions and equations
Matem´ticas 1o E.S.O.
a
Alberto Pardo Milan´s
e
-
2. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
1 Monomials
2 Adding and subtracting monomials
3 Identities and Equations
4 Solving
5 Exercises
Alberto Pardo Milan´s
e Algebraic expressions and equations
3. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Monomials
Alberto Pardo Milan´s
e Algebraic expressions and equations
4. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Monomials
What´s a monomial?
A variable is a symbol.
An algebraic expression in variables x, y, z, a, r, t . . . k is an
expression constructed with the variables and numbers using
addition, multiplication, and powers.
A number multiplied with a variable in an algebraic expression is
named coefficient.
A product of positive integer powers of a fixed set of variables
multiplied by some coefficient is called a monomial.
2 2 2 3
Examples: 3x, xy , x y z.
3
Alberto Pardo Milan´s
e Algebraic expressions and equations
5. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Monomials
Like monomials and unlike monomials
In a monomial with only one variable, the power is called its order,
or sometimes its degree.
Example: Deg(5x4 )=4.
In a monomial with several variables, the order/degree is the sum
of the powers.
Example: Deg(x2 z 4 )=6.
Monomials are called similar or like ones, if they are identical or
differed only by coefficients.
2
Example: 2x3 y 2 and x3 y 2 are like monomials. 4xy 2 and 4y 2 x4
5
are unlike monomials.
Alberto Pardo Milan´s
e Algebraic expressions and equations
6. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Adding and subtracting
monomials
Alberto Pardo Milan´s
e Algebraic expressions and equations
7. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Adding and subtracting monomials
Adding and Subtracting
You can ONLY add or subtract like monomials.
To add or subtract like monomials use the same rules as with
integers.
Example: 3x + 4x = (3 + 4)x = 7x.
Example: 20a − 24a = (20 − 24)a = −4a.
Example: 7x + 5y ⇐= you can´t add unlike monomials.
Alberto Pardo Milan´s
e Algebraic expressions and equations
8. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Identities and Equations
Alberto Pardo Milan´s
e Algebraic expressions and equations
9. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Identities and Equations
What´s an equation? Identities vs equations.
An equation is a mathematical expression stating that a pair of
algebraic expression are the same.
If the equation is true for every value of the variables then it´s
called Identity.
An identity is a mathematical relationship equating one quantity to
another which may initially appear to be different.
Example: x2 − x3 + x + 1 = 3x4 is an equation,
3x2 − x + 1 = x2 − x + 2 + 2x2 − 1 is an identity.
Alberto Pardo Milan´s
e Algebraic expressions and equations
10. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Identities and Equations
Parts of an equation.
In an equation:
the variables are named unknowns (or indeterminate quantities),
the number multiplied with a variable is named coefficient, a
term is a summand of the equation, the highest power of the
unknowns is called the order/degree of the equation.
Example: In the equation 2x3 + 4y + 1 = 4:
the unknowns are x and y,
the coefficient of x3 is 2
and
the coefficient of y is 4,
the order of the equation is 3.
Alberto Pardo Milan´s
e Algebraic expressions and equations
11. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Identities and Equations
Parts of an equation.
In an equation:
the variables are named unknowns (or indeterminate quantities),
the number multiplied with a variable is named coefficient, a
term is a summand of the equation, the highest power of the
unknowns is called the order/degree of the equation.
Example: In the equation 2x3 + 4y + 1 = 4:
the unknowns are x and y,
the coefficient of x3 is 2
and
the coefficient of y is 4,
the order of the equation is 3.
Alberto Pardo Milan´s
e Algebraic expressions and equations
12. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Identities and Equations
Parts of an equation.
In an equation:
the variables are named unknowns (or indeterminate quantities),
the number multiplied with a variable is named coefficient, a
term is a summand of the equation, the highest power of the
unknowns is called the order/degree of the equation.
Example: In the equation 2x3 + 4y + 1 = 4:
the unknowns are x and y,
the coefficient of x3 is 2
and
the coefficient of y is 4,
the order of the equation is 3.
Alberto Pardo Milan´s
e Algebraic expressions and equations
13. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Identities and Equations
Parts of an equation.
In an equation:
the variables are named unknowns (or indeterminate quantities),
the number multiplied with a variable is named coefficient, a
term is a summand of the equation, the highest power of the
unknowns is called the order/degree of the equation.
Example: In the equation 2x3 + 4y + 1 = 4:
the unknowns are x and y,
the coefficient of x3 is 2
and
the coefficient of y is 4,
the order of the equation is 3.
Alberto Pardo Milan´s
e Algebraic expressions and equations
14. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Identities and Equations
Parts of an equation.
In an equation:
the variables are named unknowns (or indeterminate quantities),
the number multiplied with a variable is named coefficient, a
term is a summand of the equation, the highest power of the
unknowns is called the order/degree of the equation.
Example: In the equation 2x3 + 4y + 1 = 4:
the unknowns are x and y,
the coefficient of x3 is 2
and
the coefficient of y is 4,
the order of the equation is 3.
Alberto Pardo Milan´s
e Algebraic expressions and equations
15. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
Alberto Pardo Milan´s
e Algebraic expressions and equations
16. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
Solution of an equation.
You are solving a equation when you replace a variable with a
value and the mathematical expressions are still the same. The
value for the variables is the solution of the equation.
Example: In the equation 2x = 10 the solution is 5, because
2 · 5 = 10.
Example: Sam is 9 years old. This is seven years younger than her
sister Rose’s age. We can solve an equation to find Rose’s age:
x − 7 = 9, the solution of the equation is 16, so Rose is 16 years
old.
Alberto Pardo Milan´s
e Algebraic expressions and equations
17. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
Solution of an equation.
You are solving a equation when you replace a variable with a
value and the mathematical expressions are still the same. The
value for the variables is the solution of the equation.
Example: In the equation 2x = 10 the solution is 5, because
2 · 5 = 10.
Example: Sam is 9 years old. This is seven years younger than her
sister Rose’s age. We can solve an equation to find Rose’s age:
x − 7 = 9, the solution of the equation is 16, so Rose is 16 years
old.
Alberto Pardo Milan´s
e Algebraic expressions and equations
18. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
Solution of an equation.
You are solving a equation when you replace a variable with a
value and the mathematical expressions are still the same. The
value for the variables is the solution of the equation.
Example: In the equation 2x = 10 the solution is 5, because
2 · 5 = 10.
Example: Sam is 9 years old. This is seven years younger than her
sister Rose’s age. We can solve an equation to find Rose’s age:
x − 7 = 9, the solution of the equation is 16, so Rose is 16 years
old.
Alberto Pardo Milan´s
e Algebraic expressions and equations
19. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
To solve equations you can use the balance method, you must
carry out the same operations in both sides and in the same order.
You must use these properties:
• Addition Property of Equalities: If you add the same number to
each side of an equation, the two sides remain equal (note you can
also add negative numbers).
Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2
• Multiplication Property of Equalities: If you multiply by the same
number each side of an equation, the two sides remain equal (note
you can also multiply by fractions).
x x
Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30
5 5
Alberto Pardo Milan´s
e Algebraic expressions and equations
20. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
To solve equations you can use the balance method, you must
carry out the same operations in both sides and in the same order.
You must use these properties:
• Addition Property of Equalities: If you add the same number to
each side of an equation, the two sides remain equal (note you can
also add negative numbers).
Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2
• Multiplication Property of Equalities: If you multiply by the same
number each side of an equation, the two sides remain equal (note
you can also multiply by fractions).
x x
Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30
5 5
Alberto Pardo Milan´s
e Algebraic expressions and equations
21. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
To solve equations you can use the balance method, you must
carry out the same operations in both sides and in the same order.
You must use these properties:
• Addition Property of Equalities: If you add the same number to
each side of an equation, the two sides remain equal (note you can
also add negative numbers).
Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2
• Multiplication Property of Equalities: If you multiply by the same
number each side of an equation, the two sides remain equal (note
you can also multiply by fractions).
x x
Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30
5 5
Alberto Pardo Milan´s
e Algebraic expressions and equations
22. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
To solve equations you can use the balance method, you must
carry out the same operations in both sides and in the same order.
You must use these properties:
• Addition Property of Equalities: If you add the same number to
each side of an equation, the two sides remain equal (note you can
also add negative numbers).
Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2
• Multiplication Property of Equalities: If you multiply by the same
number each side of an equation, the two sides remain equal (note
you can also multiply by fractions).
x x
Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30
5 5
Alberto Pardo Milan´s
e Algebraic expressions and equations
23. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
• Brackets: Sometimes you will need to solve equations involving
brackets. If brackets appear, first remove the brackets by expanding
each bracketed expression.
Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
2x 8
2x = 8 =⇒ = =⇒ x = 4
2 2
Use all three properties to solve equations:
Example: Solve 4x + 3 · (x − 25) = 240:
First we remove brackets: 3 · (x − 25) = 3x − 75 so
4x+3x − 75 = 240.
Them we use addition property:
4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
7x 315
Now we can use multiplication property: =
7 7
So the solution is x = 45.
Alberto Pardo Milan´s
e Algebraic expressions and equations
24. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
• Brackets: Sometimes you will need to solve equations involving
brackets. If brackets appear, first remove the brackets by expanding
each bracketed expression.
Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
2x 8
2x = 8 =⇒ = =⇒ x = 4
2 2
Use all three properties to solve equations:
Example: Solve 4x + 3 · (x − 25) = 240:
First we remove brackets: 3 · (x − 25) = 3x − 75 so
4x+3x − 75 = 240.
Them we use addition property:
4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
7x 315
Now we can use multiplication property: =
7 7
So the solution is x = 45.
Alberto Pardo Milan´s
e Algebraic expressions and equations
25. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
• Brackets: Sometimes you will need to solve equations involving
brackets. If brackets appear, first remove the brackets by expanding
each bracketed expression.
Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
2x 8
2x = 8 =⇒ = =⇒ x = 4
2 2
Use all three properties to solve equations:
Example: Solve 4x + 3 · (x − 25) = 240:
First we remove brackets: 3 · (x − 25) = 3x − 75 so
4x+3x − 75 = 240.
Them we use addition property:
4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
7x 315
Now we can use multiplication property: =
7 7
So the solution is x = 45.
Alberto Pardo Milan´s
e Algebraic expressions and equations
26. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
• Brackets: Sometimes you will need to solve equations involving
brackets. If brackets appear, first remove the brackets by expanding
each bracketed expression.
Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
2x 8
2x = 8 =⇒ = =⇒ x = 4
2 2
Use all three properties to solve equations:
Example: Solve 4x + 3 · (x − 25) = 240:
First we remove brackets: 3 · (x − 25) = 3x − 75 so
4x+3x − 75 = 240.
Them we use addition property:
4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
7x 315
Now we can use multiplication property: =
7 7
So the solution is x = 45.
Alberto Pardo Milan´s
e Algebraic expressions and equations
27. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
• Brackets: Sometimes you will need to solve equations involving
brackets. If brackets appear, first remove the brackets by expanding
each bracketed expression.
Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
2x 8
2x = 8 =⇒ = =⇒ x = 4
2 2
Use all three properties to solve equations:
Example: Solve 4x + 3 · (x − 25) = 240:
First we remove brackets: 3 · (x − 25) = 3x − 75 so
4x+3x − 75 = 240.
Them we use addition property:
4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
7x 315
Now we can use multiplication property: =
7 7
So the solution is x = 45.
Alberto Pardo Milan´s
e Algebraic expressions and equations
28. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
• Brackets: Sometimes you will need to solve equations involving
brackets. If brackets appear, first remove the brackets by expanding
each bracketed expression.
Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
2x 8
2x = 8 =⇒ = =⇒ x = 4
2 2
Use all three properties to solve equations:
Example: Solve 4x + 3 · (x − 25) = 240:
First we remove brackets: 3 · (x − 25) = 3x − 75 so
4x+3x − 75 = 240.
Them we use addition property:
4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
7x 315
Now we can use multiplication property: =
7 7
So the solution is x = 45.
Alberto Pardo Milan´s
e Algebraic expressions and equations
29. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Solving
The balance method.
• Brackets: Sometimes you will need to solve equations involving
brackets. If brackets appear, first remove the brackets by expanding
each bracketed expression.
Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
2x 8
2x = 8 =⇒ = =⇒ x = 4
2 2
Use all three properties to solve equations:
Example: Solve 4x + 3 · (x − 25) = 240:
First we remove brackets: 3 · (x − 25) = 3x − 75 so
4x+3x − 75 = 240.
Them we use addition property:
4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
7x 315
Now we can use multiplication property: =
7 7
So the solution is x = 45.
Alberto Pardo Milan´s
e Algebraic expressions and equations
30. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Alberto Pardo Milan´s
e Algebraic expressions and equations
31. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 1
Solve the equations:
a 4x + 2 = 26
b 5(2x − 1) = 7(9 − x)
x 2x
c + =7
2 3
d 19 + 4x = 9 − x
e 3(2x + 1) = x − 2
x 3x 1
f − =
5 10 5
Alberto Pardo Milan´s
e Algebraic expressions and equations
32. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 1
Solve the equations:
x 2x
a 4x + 2 = 26 c + =7
2 3
4x = 26 − 2 3x 4x 42
+ =
4x = 24 6 6 6
x = 24 : 4 3x + 4x = 42
x=6 7x = 42
x = 42 : 7
b 5(2x − 1) = 7(9 − x)
x=6
10x − 5 = 63 − 7x
10x + 7x = 63 + 5 d 19 + 4x = 9 − x
17x = 68 4x + x = 9 − 19
x = 68 : 17 5x = −10
x=4 x = −10 : 5
x = −2
Alberto Pardo Milan´s
e Algebraic expressions and equations
33. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 1
Solve the equations:
x 2x
a 4x + 2 = 26 c + =7
2 3
4x = 26 − 2 3x 4x 42
+ =
4x = 24 6 6 6
x = 24 : 4 3x + 4x = 42
x=6 7x = 42
x = 42 : 7
b 5(2x − 1) = 7(9 − x)
x=6
10x − 5 = 63 − 7x
10x + 7x = 63 + 5 d 19 + 4x = 9 − x
17x = 68 4x + x = 9 − 19
x = 68 : 17 5x = −10
x=4 x = −10 : 5
x = −2
Alberto Pardo Milan´s
e Algebraic expressions and equations
34. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 1
Solve the equations:
x 2x
a 4x + 2 = 26 c + =7
2 3
4x = 26 − 2 3x 4x 42
+ =
4x = 24 6 6 6
x = 24 : 4 3x + 4x = 42
x=6 7x = 42
x = 42 : 7
b 5(2x − 1) = 7(9 − x)
x=6
10x − 5 = 63 − 7x
10x + 7x = 63 + 5 d 19 + 4x = 9 − x
17x = 68 4x + x = 9 − 19
x = 68 : 17 5x = −10
x=4 x = −10 : 5
x = −2
Alberto Pardo Milan´s
e Algebraic expressions and equations
35. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 1
Solve the equations:
x 2x
a 4x + 2 = 26 c + =7
2 3
4x = 26 − 2 3x 4x 42
+ =
4x = 24 6 6 6
x = 24 : 4 3x + 4x = 42
x=6 7x = 42
x = 42 : 7
b 5(2x − 1) = 7(9 − x)
x=6
10x − 5 = 63 − 7x
10x + 7x = 63 + 5 d 19 + 4x = 9 − x
17x = 68 4x + x = 9 − 19
x = 68 : 17 5x = −10
x=4 x = −10 : 5
x = −2
Alberto Pardo Milan´s
e Algebraic expressions and equations
36. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 1
Solve the equations:
e 3(2x + 1) = x − 2
6x + 3 = x − 2
6x − x = −2 − 3
5x = −5
x = −5 : 5
x = −1
x 3x 1
f − =
5 10 5
2x 3x 2
− =
10 10 10
2x − 3x = 2
−1x = 2
x = 2 : (−1)
x = −2
Alberto Pardo Milan´s
e Algebraic expressions and equations
37. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 1
Solve the equations:
e 3(2x + 1) = x − 2
6x + 3 = x − 2
6x − x = −2 − 3
5x = −5
x = −5 : 5
x = −1
x 3x 1
f − =
5 10 5
2x 3x 2
− =
10 10 10
2x − 3x = 2
−1x = 2
x = 2 : (−1)
x = −2
Alberto Pardo Milan´s
e Algebraic expressions and equations
38. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 2
Find a number such that 2 less than three times the number is 10.
Alberto Pardo Milan´s
e Algebraic expressions and equations
39. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 2
Find a number such that 2 less than three times the number is 10.
Data: Let x be the number.
Three times the number is 3x.
2 less than three times the number is (3x − 2) and this is
10.
3x − 2 = 10
3x = 12
x = 12 : 3
x=4 Answer: The number is
x = 4.
Alberto Pardo Milan´s
e Algebraic expressions and equations
40. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 2
Find a number such that 2 less than three times the number is 10.
Data: Let x be the number.
Three times the number is 3x.
2 less than three times the number is (3x − 2) and this is
10.
3x − 2 = 10
3x = 12
x = 12 : 3
x=4 Answer: The number is
x = 4.
Alberto Pardo Milan´s
e Algebraic expressions and equations
41. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 2
Find a number such that 2 less than three times the number is 10.
Data: Let x be the number.
Three times the number is 3x.
2 less than three times the number is (3x − 2) and this is
10.
3x − 2 = 10
3x = 12
x = 12 : 3
x=4 Answer: The number is
x = 4.
Alberto Pardo Milan´s
e Algebraic expressions and equations
42. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 3
Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155
pounds less than twice her husband’s money. How many pounds
does Mr. Roberts have? How many pounds does Mrs. Roberts
have?
Alberto Pardo Milan´s
e Algebraic expressions and equations
43. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 3
Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155
pounds less than twice her husband’s money. How many pounds
does Mr. Roberts have? How many pounds does Mrs. Roberts
have?
Data: They have 370 pounds. Mr. Roberts has x pounds.
Twice Mr. Roberts’ money is 2x pounds.
Mrs. Roberts has (2x − 155) pounds.
x+(2x−155) = 370
x + 2x − 155 = 370
3x = 525
x = 525 : 3
x = 175 Answer: Mr. Roberts has 175
370 − 175 = 195 pounds. Mrs. Roberts has 195
pounds.
Alberto Pardo Milan´s
e Algebraic expressions and equations
44. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 3
Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155
pounds less than twice her husband’s money. How many pounds
does Mr. Roberts have? How many pounds does Mrs. Roberts
have?
Data: They have 370 pounds. Mr. Roberts has x pounds.
Twice Mr. Roberts’ money is 2x pounds.
Mrs. Roberts has (2x − 155) pounds.
x+(2x−155) = 370
x + 2x − 155 = 370
3x = 525
x = 525 : 3
x = 175 Answer: Mr. Roberts has 175
370 − 175 = 195 pounds. Mrs. Roberts has 195
pounds.
Alberto Pardo Milan´s
e Algebraic expressions and equations
45. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 3
Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155
pounds less than twice her husband’s money. How many pounds
does Mr. Roberts have? How many pounds does Mrs. Roberts
have?
Data: They have 370 pounds. Mr. Roberts has x pounds.
Twice Mr. Roberts’ money is 2x pounds.
Mrs. Roberts has (2x − 155) pounds.
x+(2x−155) = 370
x + 2x − 155 = 370
3x = 525
x = 525 : 3
x = 175 Answer: Mr. Roberts has 175
370 − 175 = 195 pounds. Mrs. Roberts has 195
pounds.
Alberto Pardo Milan´s
e Algebraic expressions and equations
46. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 4
The length of a room exceeds the width by 5 feet. The length of
the four walls is 30 feet. Find the dimensions of the room.
Alberto Pardo Milan´s
e Algebraic expressions and equations
47. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 4
The length of a room exceeds the width by 5 feet. The length of
the four walls is 30 feet. Find the dimensions of the room.
Data: The width of the room is x feet.
The length of the room is (x + 5) feet.
The length of the four walls is x + (x + 5) + x + (x + 5) feet
and this is 30 feet.
x + (x + 5) + x + (x + 5) = 30
x + x + 5 + x + x + 5 = 30
x + x + x + x = 30 − 5 − 5
4x = 20
x = 20 : 4
x=5
5 + 5 = 10 Answer: The room is 5 feet
long and 10 feet wide.
Alberto Pardo Milan´s
e Algebraic expressions and equations
48. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 4
The length of a room exceeds the width by 5 feet. The length of
the four walls is 30 feet. Find the dimensions of the room.
Data: The width of the room is x feet.
The length of the room is (x + 5) feet.
The length of the four walls is x + (x + 5) + x + (x + 5) feet
and this is 30 feet.
x + (x + 5) + x + (x + 5) = 30
x + x + 5 + x + x + 5 = 30
x + x + x + x = 30 − 5 − 5
4x = 20
x = 20 : 4
x=5
5 + 5 = 10 Answer: The room is 5 feet
long and 10 feet wide.
Alberto Pardo Milan´s
e Algebraic expressions and equations
49. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 4
The length of a room exceeds the width by 5 feet. The length of
the four walls is 30 feet. Find the dimensions of the room.
Data: The width of the room is x feet.
The length of the room is (x + 5) feet.
The length of the four walls is x + (x + 5) + x + (x + 5) feet
and this is 30 feet.
x + (x + 5) + x + (x + 5) = 30
x + x + 5 + x + x + 5 = 30
x + x + x + x = 30 − 5 − 5
4x = 20
x = 20 : 4
x=5
5 + 5 = 10 Answer: The room is 5 feet
long and 10 feet wide.
Alberto Pardo Milan´s
e Algebraic expressions and equations
50. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 5
Maria spent a third of her money on food. Then, she spent e21 on
a present. At the end, she had the fifth of her money. How much
money did she have at the beginning?
Alberto Pardo Milan´s
e Algebraic expressions and equations
51. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 5
Maria spent a third of her money on food. Then, she spent e21 on
a present. At the end, she had the fifth of her money. How much
money did she have at the beginning?
x
Data: At the beginig She had x euros. She spent on food
3
x
and e21 on a present. At the end She had euros.
5
x x
x − − 21 = 7x = 315
3 5
x x 21 x x = 315 : 7
− − = x = 45
1 3 1 5
15x 5x 315 3x
− − =
15 15 15 15 Answer: At the beginig
15x − 5x − 315 = 3x She had e45.
15x − 5x − 3x = 315
Alberto Pardo Milan´s
e Algebraic expressions and equations
52. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 5
Maria spent a third of her money on food. Then, she spent e21 on
a present. At the end, she had the fifth of her money. How much
money did she have at the beginning?
x
Data: At the beginig She had x euros. She spent on food
3
x
and e21 on a present. At the end She had euros.
5
x x
x − − 21 = 7x = 315
3 5
x x 21 x x = 315 : 7
− − = x = 45
1 3 1 5
15x 5x 315 3x
− − =
15 15 15 15 Answer: At the beginig
15x − 5x − 315 = 3x She had e45.
15x − 5x − 3x = 315
Alberto Pardo Milan´s
e Algebraic expressions and equations
53. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 5
Maria spent a third of her money on food. Then, she spent e21 on
a present. At the end, she had the fifth of her money. How much
money did she have at the beginning?
x
Data: At the beginig She had x euros. She spent on food
3
x
and e21 on a present. At the end She had euros.
5
x x
x − − 21 = 7x = 315
3 5
x x 21 x x = 315 : 7
− − = x = 45
1 3 1 5
15x 5x 315 3x
− − =
15 15 15 15 Answer: At the beginig
15x − 5x − 315 = 3x She had e45.
15x − 5x − 3x = 315
Alberto Pardo Milan´s
e Algebraic expressions and equations
54. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 6
John bought a book, a pencil and a notebook. The book cost the
double of the notebook, and the pencil cost the fifth of the book
and the notebook together. If he paid e18, what is the price of
each article?
Alberto Pardo Milan´s
e Algebraic expressions and equations
55. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 6
John bought a book, a pencil and a notebook. The book cost the
double of the notebook, and the pencil cost the fifth of the book
and the notebook together. If he paid e18, what is the price of
each article?
Data: The notebook cost x euros. The book cost 2x euros.
The notebook and the book together cost (x + 2x) euros.
x + 2x
The pencil cost . He paid e18.
5
x + 2x 5x+10x+x+2x = 90
x + 2x + = 18
5 18x = 90
x 2x x + 2x 18 x = 90 : 18
+ + =
1 1 5 1 x=5
5x 10x x + 2x 90 2x = 10
+ + = x + 2x 15
5 5 5 5 = = 3.
5x + 10x + (x + 2x) = 90 5 5
Alberto Pardo Milan´s
e Algebraic expressions and equations
56. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 6
Data: The notebook cost x euros. The book cost 2x euros.
The notebook and the book together cost (x + 2x) euros.
x + 2x
The pencil cost . He paid e18.
5
x + 2x 5x+10x+x+2x = 90
x + 2x + = 18
5 18x = 90
x 2x x + 2x 18 x = 90 : 18
+ + =
1 1 5 1 x=5
5x 10x x + 2x 90 2x = 10
+ + = x + 2x 15
5 5 5 5 = = 3.
5x + 10x + (x + 2x) = 90 5 5
Answer: The notebook cost e5, the book e10 and the
pencil e3 .
Alberto Pardo Milan´s
e Algebraic expressions and equations
57. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises
Exercises
Exercise 6
Data: The notebook cost x euros. The book cost 2x euros.
The notebook and the book together cost (x + 2x) euros.
x + 2x
The pencil cost . He paid e18.
5
x + 2x 5x+10x+x+2x = 90
x + 2x + = 18
5 18x = 90
x 2x x + 2x 18 x = 90 : 18
+ + =
1 1 5 1 x=5
5x 10x x + 2x 90 2x = 10
+ + = x + 2x 15
5 5 5 5 = = 3.
5x + 10x + (x + 2x) = 90 5 5
Answer: The notebook cost e5, the book e10 and the
pencil e3 .
Alberto Pardo Milan´s
e Algebraic expressions and equations