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Mathematics 8.

Table of contents
Algebra
Algebraic expressions (revision)
How to solve equations and inequalities? (revision)
Multiplication of algebraic expressions formed by multiple terms
The square of a sum or a difference (supplementary material)
The product of a sum and a difference (supplementary material)
Factorisation
Algebraic fractions (supplementary material)
Solving equations by factorisation
Mixed exercises
Word problems
Sets, combinatorics
Geometry I.
Three-dimensional geometry
Statistics, probability
Geometry II.
Functions, sequences
Algebraic expressions (revision)
Algebraic expressions are used to represent general mathematical relationships or to write computer programs. In these expressions, numbers and letters are connected by operators and parentheses. The letters, also known as variables, can be replaced by numbers that are elements of the given domain.
Example 1
A family of four is going on holiday and takes out travel insurance for \latex{ seven}-\latex{ days } from the True Travel insurance company. The insurance costs €\latex{ 2 } \latex{ per } \latex{ person } \latex{ per } \latex{ day. } 
a) How much does the travel insurance cost for the whole family?
b) Pete compares the offers of different insurance companies. He fills a table with the collected data and writes an algebraic expression that a program can use to calculate the insurance costs. Write down this algebraic expression if the holiday lasts for \latex{ n } \latex{ days } and the insurance costs \latex{ b } \latex{ euros } \latex{ per } \latex{ person } \latex{ per } \latex{ day. } 
c) True Travel offers a \latex{15}% family discount. Calculate the reduced price and express it using an algebraic expression. 
d) Joyful Journey offers \latex{ seven }-\latex{ day } travel insurance at €\latex{ 1.8 } \latex{ per } \latex{ person } \latex{ per } \latex{ day. } If the company also offers a \latex{ 15 }% family discount, how much should a family of four pay for a \latex{ one }-\latex{ week } holiday?
Solution
a) The cost of the insurance
for \latex{ one\, person } for \latex{ one } \latex{day} is \latex{ 2\,euros}
for \latex{ one\, person } for \latex{ seven } \latex{days}: \latex{7\times 2=}\latex{14}
for \latex{ four } \latex{ people } for \latex{ seven } \latex{days}: \latex{4\times 7\times 2=}\latex{ 56 }
 

Answer: The travel insurance costs €\latex{ 56 } for a family of four for seven \latex{ days. }

 

b) The cost of the insurance

for \latex{ one } \latex{ person } for \latex{ one } \latex{day} is \latex{b} \latex{ euros }

for \latex{ one } \latex{ person } for \latex{n} \latex{days}: \latex{n\times b=nb}

for \latex{ four } \latex{ people } for \latex{n} \latex{days}: \latex{4\times n\times b=4nb}

 

Answer: The cost of the insurance for a family of four can be calculated using the algebraic expression \latex{ 4nb }.

 
c) Write down the cost of the family insurance both in numbers and as an algebraic expression. 
The discount
algebraic expression
The cost of the insurance
The remaining amount
\latex{ 4 \times n \times x = 4nb }
calculation
\latex{ 56 }
\latex{ 0.15 \times 56 = 8.4}
\latex{ 56 - 8.4 = 47.6}
\latex{ 4nb - 0.6nb = 3.4nb}
\latex{0.15 \times 4nb = 0.6nb}

Answer: The cost of the insurance with the discount is €\latex{ 47.6 }, which can be represented by the algebraic expression \latex{ 3.4 nb} \latex{euros}.

 

d) The offer from Joyful Journey differs from that of True Travel only in the daily price. Therefore, the algebraic expression \latex{ 3.4nb } can be used for the calculation.

In this expression, \latex{ n =7 } and \latex{ b = 1.8 }.

 

                             \latex{3.4 \,nb = 3.4 \times 7 \times 1.8 = 42.84}

 

Answer: The family should pay €\latex{42.84} at Joyful Journey.

 

 

 

 

 

 

 

The algebraic expression \latex{4\times n\times b} is a single-term expression (monomial), because multiplication is the last operation to be performed.

In the algebraic expression \latex{4\times n\times b}, the coefficient of \latex{n\times b} is \latex{4}.

 

The algebraic expression \latex{ 4nb - 0.6nb } is a polynomial expression with two terms, because the last operation to be performed is subtraction. The terms \latex{ 4nb } and \latex{ 0.6nb } are like terms because they differ only in their numerical coefficients. Therefore, the coefficient of their difference is equal to the difference between their coefficients.

When you replace the letters in the algebraic expression with numerical values from the domain, you perform substitution. 
Example 2
The prime factorisation of  \latex{ 2009 } is \latex{2009=7^{2}\times 41 }, which can be written in the form \latex{ p^{2} \times q }, where \latex{ p } and \latex{ q } are distinct prime numbers. Which is the first \latex{ year } after \latex{ 2009 } that can be represented by the same expression?
\latex{ p^2 = p \times p }
Solution
Write down the prime factorisation of the following \latex{ years. } 
 
\latex{ 2010 = 2\times 3 \times 5 \times 67 }
\latex{ 2011 } is a prime number
\latex{ 2012 = 2^2 \times 503 }, where \latex{ 503 } is a prime number
 
Answer: Therefore, the first \latex{ year } after \latex{ 2009 }, which can be represented by the same algebraic expression, is \latex{ 2012 }.
Try to write the current \latex{ year } in a similar form. Then, find the next \latex{ year } that can be written in the same way.
Example 3
Match the equivalent algebraic expressions in the two columns.
\latex{ 2x^3 + 4x^2}
a)
b)
c)
d)
e)
V.
IV.
III.
II.
I.
\latex{ 2x +5x }
\latex{ 3x^3y }
\latex{ 5a\times a\times 2a }
\latex{ 3.5a^2+3.5ab }
\latex{ 2x (x \times x + 2x) }
\latex{ 7x }
\latex{ 3xy \times x^2 }
\latex{ \frac{7}{2}a \times(a+b) }
\latex{ 10a^3}
Solution
First, examine which algebraic expressions can be simplified by performing mathematical operations.

a) In the expressions that are formed by a group of terms, the like terms can be combined. 
\latex{ 2x + 5x =7x }

c)  A product can be rewritten as a power. 
\latex{ 5a \times a \times 2a =5 \times 2 \times a \times a \times a=10a^3 }

IV. Exponential terms with the same base can be multiplied.
\latex{ 3xy \times x^2 =3x \times x^2 \times y =3x^3 \times y }

In algebraic expression e) and III. the multiplication can be converted into an addition.
\latex{ \frac{7}{2}a \times (a+b) =3.5a\times(a + b) =3.5a\times a + 3.5a\times b = 3.5a^2 + 3.5ab \\[5pt]}.
\latex{ 2x(x \times x+ 2x) =2x(x^2 + 2x) =2x \times x^2+ 2x \times 2x =2x^3 + 4x^2 }
 
Thus, the equivalent algebraic expressions are as follows:

 

 

 

 

 

 

 

 

 

\latex{ a \times a \times a = a^3 }

 

 

 

\latex{ x \times x^2 = x^1 \times x^2 = x^3 }

 

 

 

When multiplying a sum by a number, multiply each term of the sum by that number, then add the resulting products.

\latex{ 2x^3 + 4x^2}
a)
b)
c)
d)
e)
V.
IV.
III.
II.
I.
\latex{ 2x +5x }
\latex{ 3x^3y }
\latex{ 5a\times a\times 2a }
\latex{ 3.5a^2+3.5ab }
\latex{ 2x (x \times x + 2x) }
\latex{ 7x }
\latex{ 3xy \times x^2 }
\latex{ \frac{7}{2}a \times(a+b) }
\latex{ 10a^3}
Example 4
Perform the following divisions.
a) \latex{ \frac{10a \times 14a}{2} }
b) \latex{ \frac{9a \times 6b}{3} }
c) \latex{ \frac{10a + 14a}{2} }
d) \latex{ \frac{9a + 6b}{3} }
Solution 
a) \latex{ \frac{\overset5{\bcancel{10}} a \times 14a}{\underset1{\bcancel2}} =70a^2}
or
\latex{ \frac{10a × \overset{7}{\bcancel{14}} a}{\underset1{\bcancel2}} =70a^2}
d) \latex{ \frac{\overset{3}{\bcancel{9}} a \times 6b}{\underset{1}{\bcancel{3}} }=18ab }
or
\latex{ \frac{9a × \overset{2}{\bcancel{6}} b}{\underset{1}{\bcancel{3}} }=18ab }
When dividing a product by a number, divide one of the factors by that number and then multiply the result by the other factor.
c) In the numerator of the fraction, the terms can be combined:
 
\latex{ \frac{10a + 14a}{2} = \frac{24a}{2} = 12a}
 
b) Each term of the sum is divided by the denominator, and then the quotients are added together:
 
\latex{ \frac{9a + 6b}{3} = \frac{9}{3}a + \frac{6}{3} b = 3a +2b}
Example 5
Simplify the following algebraic expressions.
a) \latex{3 \times (2x - 1) + (3 - x) \times 5}
b) \latex{y \times (y + 1) - y \times (y - 1)}
c) \latex{ \frac{x}{3} + \frac{x}{4}-\frac{x}{5} }
d) \latex{ \frac{x}{2} + \frac{x-1}{3} }
e) \latex{ \frac{5a-b}{6} + \frac{a-2b}{3} }
Solution
a) Perform the multiplications, then combine like terms.
\latex{ 3 \times (2x - 1) \;}\latex{+}\latex{\; (3 - x) \times 5 = 3 \times 2x - 3 \times 1 \;}\latex{+}\latex{\; 3 \times 5\;}\latex{ -}\latex{\;x \times 5 =} \latex{= 6x - 3 + 15 - 5x = 6x - 5x - 3 + 15 = x + 12 }
 
b)  \latex{ y \times (y + 1)\;}\latex{ -}\latex{\; y \times (y - 1) = y \times y + y \times 1\;}\latex{ -}\latex{\; y \times y\;}\latex{ +}\latex{\; y \times 1 =}
      \latex{= y^2 + y - y^2 + y = y^2 - y^2 + y + y = 2y }
 
c) Express the fractions with a common denominator.
\latex{ \frac{x}{3} + \frac{x}{4}-\frac{x}{5} = \frac{20x}{60} + \frac{15x}{60}-\frac{12x}{60} = \frac{20x+15x-12x}{60} = \frac{23x}{60}}

d) Express the fractions with a common denominator.
\latex{ \frac{x}{2}+\frac{x-1}{3}=\frac{3x+2\times (x-1)}{6}=\frac{3x+2x-2}{6}=\frac{5x-2}{6} }
When rewriting the fraction with a common denominator, the sum in the numerator should be placed in parentheses.
 
e)  Rewrite the fractions with a common denominator, then combine like terms.
\latex{ \frac{5a-b}{6} - \frac{a-2b}{3} = \frac{5a-b}{6} - \frac{2\times(a-2b)}{2\times3} = \frac{5a-b-2\times(a-2b)}{6} =}
\latex{= \frac{5a-b-2a+4b}{6} = \frac{3a+3b}{6} = \frac{\overset{{1}}{\bcancel3} \times (a+b)}{\underset{2}{\bcancel{6}} } = \frac{a+b}{2} }

 

 

The sign of a product is determined by the signs of its factors.

\latex{ -y \times (-1) = +y }
Exercises
{{exercise_number}}. Translate the following word phrases into algebraic expressions. Which of the resulting expressions are formed by a single term and which by a group of terms?
  1. Five times the sum of \latex{ x } and \latex{ y. }
  2. The difference between five times \latex{ x } and \latex{ y }. 
  3. The difference between the number that is \latex{7} more than \latex{ x } and \latex{ y. }
  4. The product of the number that is \latex{ 40 }% less than \latex{ x } and the number that is \latex{ 40 }% more than \latex{ y. }
  5. The number that is \latex{ 10 }% more than the sum of \latex{ x } and \latex{ y. }
{{exercise_number}}. The price of a plane ticket is €\latex{\large{a}}, and a landing fee of €\latex{\large{b}} must also be paid for each ticket.
  1. How much do you have to pay in total for \latex{ n } number of tickets, including the landing fees?
  2. Due to rising fuel prices, the price of the ticket has been increased by \latex{ 10 }%, while the landing fee has remained the same. How much do you have to pay now for \latex{ n } number of tickets, including the landing fees? 
  3. How much more do you have to pay for \latex{ n } number of tickets if both the landing fee and the price of the ticket have been increased by \latex{ 10 }%?
  4. If you buy at least \latex{5} tickets, then you do not have to pay the landing fee for \latex{ k } number of tickets, where \latex{ k \lt 5 }. How much do you have to pay in total for \latex{ n } tickets if \latex{ n\geq5 ?}
{{exercise_number}}. Perform the following divisions. 
  1. \latex{ \frac{6x\times10x}{4} }
  1. \latex{ \frac{6x+10x}{4} }
  1. \latex{ \frac{9xy+21xy}{3} }
  1. \latex{ \frac{9xy \times 21xy}{3} }
{{exercise_number}}. Simplify the following algebraic expressions.
  1. \latex{ 4a+6a-12a+7a }
  1. \latex{ -5b+3b-9b + 2b }
  1. \latex{ 11c-3c-5c-3c }
  1. \latex{-9d+10d-5d+6d}
{{exercise_number}}. Simplify the following algebraic expressions.
  1. \latex{ 5x-4+2x-7-6x+9 }
  1. \latex{ 4-5y-5+2y+2-6y }
  1. \latex{ 6v+2c-3v-1-7v+7 }
  1. \latex{ 10-5z+4+4z-15-2z }
{{exercise_number}}. Simplify the following algebraic expressions.
  1. \latex{ 5a+3b-4a-4b+2 }
  1. \latex{ -3c+2d+3-5c+2d-9 }
  1. \latex{ 2e-4f+9-7e-4f+3 }
  1. \latex{ -5g-9h-6-7g-8-5h }
{{exercise_number}}. Combine the like terms where possible.
  1. \latex{ 3a+2b+(3a-2b) }
  1. \latex{ 3a+2b-(3a-2b) }
  1. \latex{ 3a-2b-(3a-2b) }
  1. \latex{ -3a-2b-(3a+2b) }
{{exercise_number}}. Combine the like terms where possible.
  1. \latex{ (2x+5y)+(4x-3y)-(2x-3y) }
  1. \latex{ 2x+(5y-4x)-(3y-2x)-3y }
  1. \latex{ -2x-(5y+4x)-(3y-2x+3y) }
  1. \latex{ 2x-(5y-4x-3y)+(2x+3y) }
{{exercise_number}}. Perform the marked multiplications.
  1. \latex{ 4\times(a+5) }
  1. \latex{ 5\times(2b-3) }
  1. \latex{ 2(3-c) }
  1. \latex{ 4(2-3d) }
  1. \latex{ -2(2e+3) }
  1. \latex{ -5(3f-2) }
  1. \latex{ -6(3-g) }
  1. \latex{ -3(4-2h) }
{{exercise_number}}. Perform the operations, then simplify the resulting algebraic expressions.
  1. \latex{ 5(x+1)+3(x-2) }
  1. \latex{ 6(3-2y)+2(1-4y) }
  1. \latex{ 4(3v-1)+(3-4v) }
  1. \latex{ 2(3z-2)+3(3-2z) }
{{exercise_number}}. Perform the operations, then simplify the resulting algebraic expressions.
  1. \latex{ 3(4x+1)-2 \times (3x+2) }
  1. \latex{ 4(1-4y)-3(1+y) }
  1. \latex{ -2(3v+1)+3\times (4-v) }
  1. \latex{ -5(2z-3)-2(3-5z) }
{{exercise_number}}. Perform the operations, then simplify the resulting algebraic expressions.
  1. \latex{ y(2x - 3)- y(x + 1) }
  1. \latex{ 2a(2x + 3y)- a(3x - y) }
  1. \latex{ 5x(x - y)- y(2x + y) }
  1. \latex{ -3b(4b-3a)-2(ab+5b^2) }
{{exercise_number}}. Find the values of the following algebraic expressions.
  1. \latex{ [(x- y)- (y + x)] \times 2}
  2. \latex{ a(a- b)- b(b- a)}
  3. \latex{ \frac{a-b}{2} + \frac{2b}{4}}
  4. \latex{ 2(x^2- y^2)- x(2x + 1)+ x}
  1. if \latex{ x = 0.19} and \latex{y = \frac{1}{4} }
  2. if \latex{ a = 10} and \latex{b = 0.7}
  3. if \latex{ a = 20} and \latex{b = 0.19}
  4. if \latex{ x = \frac{2}{3}} and \latex{y =-1}
{{exercise_number}}. Complete the algebraic expressions to make the equations true.
  1. \latex{ 5x^2y + \fcolorbox{black}{#eaf1fa}{\textcolor{#eaf1fa}{O}} = 11x^2y}
  1. \latex{5ab - \fcolorbox{black}{#eaf1fa}{\textcolor{#eaf1fa}{O}} +3ab = 4ab}
  1. \latex{3x^2-y^2 + \fcolorbox{black}{#eaf1fa}{\textcolor{#eaf1fa}{O}} = x^2-y^2}
  1. \latex{ 2a- \fcolorbox{black}{#eaf1fa}{\textcolor{#eaf1fa}{O}} = 10a}
  1. \latex{ \frac{xy}{2} + \fcolorbox{black}{#eaf1fa}{\textcolor{#eaf1fa}{O}} = 3xy}
  1. \latex{ 2x^2y- \frac{1}{2} yx^2 -\fcolorbox{black}{#eaf1fa}{\textcolor{#eaf1fa}{O}} = 5x^2y}
{{exercise_number}}. Bring the fractions to a common denominator, then simplify the algebraic expressions.
  1. \latex{ \frac{x}{3}-\frac{x}{4} }
  1. \latex{ \frac{y-1}{4} + \frac{y}{2} }
  1.  \latex{ \frac{a+b}{2} + \frac{a-b}{2} }
  1. \latex{ \frac{x+1}{2}-\frac{x-1}{2} }
  1. \latex{ \frac{2c+1}{4}-\frac{2c-1}{2} }
  1. \latex{ \frac{x-2y}{3}-\frac{2x-y}{2} }
{{exercise_number}}. Bring the fractions to a common denominator, then simplify the algebraic expressions.
  1. \latex{ \frac{a}{3} + \frac{2a}{5} }
  1. \latex{ \frac{3b}{4}-\frac{2b}{3} }
  1. \latex{ \frac{c}{2} + \frac{2c+3}{3} }
  1. \latex{ \frac{d+1}{2} + \frac{2d}{5} }
  1. \latex{ e -\frac{2e-1}{3} }
  1. \latex{ \frac{2f+1}{5} - \frac{1-f}{3} }
  1. \latex{ \frac{2g+3}{4} - \frac{3g+1}{2} }
  1. \latex{ \frac{3h+2}{3} - \frac{5h+4}{5} }
Quiz
Nora was asked where she lived. She replied, 'I live in Budapest, on Lizzard Street. My postal code, house number, and floor are all powers of two,  and the sum of those three numbers is \latex{ 1,092 }.' What is her address?
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