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Mathematics 8.

Table of contents
Algebra
Algebraic expressions (reminder)
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 (complementary topics)
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 can be used to represent 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 universal set.
Example 1
A family of four goes on a holiday and buys seven-day travel insurance from the True Travell insurance company. The insurance costs €\latex{ 2 } per person per 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 n days and the insurance costs b euros per person per 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 seven-day travel insurance at €\latex{ 1.8 } per person per day. If the company also offers a \latex{ 15 }% family discount, how much should a family of four pay for a one-week holiday?
The algebraic expression \latex{ 4 \times n \times b } is formed by a single term because the last operation to be performed is multiplication. In the algebraic expression \latex{ 4 \times n \times b }, the coefficient of \latex{ n \times b } is \latex{ 4 }.
Solution
a) The cost of the insurance in euros per person per day is \latex{ 2 };                     \latex{ 450 };
                                  per person per seven days is               \latex{7\times2};
                                 per four people per seven days is       \latex{4 \times7\times2=56}.

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

 

b) The cost of the insurance in euros per person per day is b;                      \latex{b };

                      per person per n days is               \latex{n\times b=nb};
                      per four people per n days is      \latex{4 \times n \times b=4nb}.
Answer: The cost of the insurance for a family of four can be calculated using the algebraic expressions \latex{ 4nb }.
 
c) Calculate the cost of the insurance and represent it with 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}.

 

d) The offer from Joyful Journey differs from that of True Travel only in the daily prices; thus, 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 42.84 euros at Joyful Journey.

The algebraic expression \latex{ 4nb - 0.6nb } is formed by a group of 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.
If you replace the letters in the algebraic expression with numerical values from the universal set, you perform substitution. 
Example 2
The prime factorisation of  \latex{ 2009 } is \latex{2009=72\times 41 }, which can be represented by the algebraic expression \latex{ p^{2} \times q }, where p and q denote different prime numbers. Which is the first 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 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.
 
The first year after \latex{ 2009 }, which can be represented by the same algebraic expression, is \latex{ 2012 }.
Try to express the current year using a similar algebraic expression. Then, find the nearest year that can be represented by the same expression.
Example 3
Match the equivalent algebraic expressions.
\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) Multiplications can be rewritten as powers. 
\latex{ 5a \times a \times 2a =5 \times 2 \times a \times a \times a=10a^3 };

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

III. In algebraic expression e), 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 };
\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 the following:

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

 

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

 

When multiplying a number by a sum, you can multiply each addend of the sum by that number, then add their products together.

\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{\overset{5}{\bcancel{10}} a \times 14a}{2} =70a^2}
or
c) \latex{ \frac{10a + \overset{7}{\bcancel{14}} a}{2} =70a^2}
d) \latex{ \frac{\overset{3}{\bcancel{9}} a \times 6b}{\underset{1}{\bcancel{3}} }=18ab }
or
d) \latex{ \frac{9a + \overset{2}{\bcancel{6}} b}{\underset{1}{\bcancel{3}} }=18ab }
When dividing a product by a number, you can divide one of the factors by that number and then multiply the quotient 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}
 
d) Each term of the addition 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) Convert the multiplication into an addition, then combine the like terms.
\latex{ 3 \times (2x - 1) + (3 - x) \times 5 = 3 \times 2x - 3 \times 1 + 3 \times 5 - x \times 5 =} \latex{= 6x - 3 + 15 - 5x = 6x - 5x - 3 + 15 = x + 12. }
 
b)  \latex{ y \times (y + 1) - y \times (y - 1) = y \times y + y \times 1 - y \times y + y \times 1 =}
      \latex{= y^2 + y - y^2 + y = y^2 - y2 + y + y = 2y. }
 
c) Expand the fractions to the 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) Expand the fractions to the 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 expanding the fraction to the common denominator, the addition in the numerator should be put into parentheses.
 
e) Expand the fractions to the common denominator, then combine the 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?
a) five times the sum of x and y;
b) the difference between five times x and y; 
c) the difference between the number that is greater than x by \latex{ 7 } and y;
d) the number that is \latex{ 40 }% less than x multiplied by the number that is \latex{ 40 }% greater than y;
e) the number that is \latex{ 10 }% greater than the sum of x and y.
{{exercise_number}}. The price of a plane ticket is x EUR, and a landing fee of b EUR must also be paid for each ticket.
a) How much do you have to pay in total for n number of plane tickets, including the landing fees?
b) Due to rising fuel prices, the price of the plane ticket has increased by \latex{ 10 }%, while the landing fee has remained the same. How much do you have to pay now for n number of plane tickets, including the landing fees? 
c) How much more do you have to pay for n number of plane tickets if both the landing fee and the price of the plane ticket have increased by \latex{ 10 }%?
d) If you buy at least five plane tickets, then you do not have to pay the landing fee for k number of tickets, where k \latex{ \lt 5 }. How much do you have to pay in total for n tickets if \latex{ n\gt =5 }?
{{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{ 10-5z+4+4z-15-2z }
{{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{ 0-5z+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 in the following algebraic expressions.
  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 in the following algebraic expressions.
  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}}. Combine the like terms in the following algebraic expressions.
  1. \latex{ 4(a+5) }
  1. \latex{ 5(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 multiplications.
  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+5b2) }
{{exercise_number}}. Perform the operations, then simplify the resulting 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. \latex{ if \, x = 0.19; y = \frac{1}{4} ; }
  2. \latex{ if \, a = 10; b = 0.7; }
  3. \latex{ if \, a = 20; b = 0.19 ; }
  4. \latex{ if \, x = \frac{2}{3} ; \, 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}}. Expand the fractions to the 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}}. Expand the fractions to the 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{ 1092 }.' What is her address?

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