Download unit 5: simultaneous equations (systems of

Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
UNIT 5: SIMULTANEOUS EQUATIONS (SYSTEMS OF EQUATIONS)
Simultaneous linear equations (Systems of linear equations):
A system of linear equations is a collection of linear equations with the same variables.
axby=c
a ' xb ' y=c '
} This is a system of two linear equations with two variables.
Solving a system of equations means finding the values of the variables that make all the equations
true at the same time.
Example:
2x−3y=5
x5y=9
} The solution to this system is
x=4 and y=1.
Number of solutions of a two-variable system of equations:
When you are solving systems, you are, graphically, finding intersection of lines. (Remember that
the graph of a linear equation, axby=c , is a straight line, and its points are the solution of the
equation).
For two-variables system, there are three possible types of solutions:
CASE 1: Independent Systems:
Example:
2x y=5
x4y=6
}
The two lines cross at exactly one point.
This point is the only solution of the
system. These systems are called
independent systems.
P(2,1)
x+4y=6
2x+y=5
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
CASE 2: Inconsistent Systems:
Example:
2x y=4
4x2y=2
}
Since parallel line never cross, the system has no
solution. These systems are called inconsistent
systems.
2x+y=4
4x+2y=2
CASE 3: Dependent systems:
Example:
2x3y=3
4x6y=6
}
The two lines are the same line. Any point of
the line is solution to the system. These
systems are called dependent systems.
2x+3y=3
4x+6y=6
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
Your
Turn
1. Use the graphical method to solve the following systems of equations:
}
a)
x y=2
x− y=−4
b)
x− y=3
2x−2y=6
}
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
}
c)
x − y=2
x−2y=3
d)
2x− y=−1
−4x2y=−2
e)
x− y=1
2x−3y=4
}
}
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
f)
x− y=5
3x y=3
Teacher: Miguel Angel Hernández Lorenzo.
}
Solving simultaneous equations:
1. The Substitution Method: Find the value of one unknown in either of the given equation
and substitute this value in the other equation.
Example:
x2y=3
2x−3y=6
}
2. The “Equating” Method: Find the values of one unknown in both equations and both
values are equal.
Example:
2x3y=3
3x−5y=14
}
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
3. The Elimination Method: It consists in adding equations together to eliminate variables.
Sometimes you have to add equations by a number before you add them. The goal is to end
up with one equation that has just one variable. Then, you can use substitution method to
find out the value of the other variable.
Example:
3x4y=17
2x5y=16
}
Your
Turn
1. Use the substitution method to solve the following systems of equations:
a)
x5y=7
3x−5y=11
b)
5x y=8
3x− y=11
}
}
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
c)
3x10y=6
x2y=1
Teacher: Miguel Angel Hernández Lorenzo.
}
2. Use the equating method to solve the following systems of equations:
}
a)
x2y=3
2x−3y=6
b)
2x y=4
4x−3y=−7
c)
5x2y=4
3x4y=8
}
}
3. Use the elimination method to solve the following systems of equations:
a)
2x3y=11
3x−2y=−3
}
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
b)
2x−5y=6
8x y =3
}
c)
5x2y=4
3x4y=8
}
Teacher: Miguel Angel Hernández Lorenzo.
4. Solve the following systems of equations:
}
a)
x y
− =4
3 2
x y
− =2
2 4
b)
x2 3y−1 −3
−
=
5
10
10
2x3 y7 19

=
8
4
8
}
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
}
x15 3 y1 y
−
=3
8
16
c)
7−x 1 y
−
=3
2
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5. The admission fee at a small fair is $1,50 for children and $4 for adults. On a certain day,
2.200 people enter the fair and $5.050 is collected. How many children and how many
adults went to the fair?
6. The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is
increased by 27. Find the number.
7. A landscaping company placed two orders with a nursery. The first order was 13 bushes and
4 trees, and totalled $487. The second order was for 6 bushes and 2 trees, and totalled $232.
The bill does not list the per-item price. What is the cost of one bush and of one tree?
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
8. A boy has a number of coins in both hands. If he passes two coins from the right hand to the
left hand, there will be the same amount of coins in both hands. If he passes three coins from
the left hand to the right hand, he will have double the amount of coins in his right hand than
in his left hand. At the beginning, how many coins did the boy have in each hand?
9. Find the ages of two people knowing that ten years ago, the age of the first person was four
times the age of the second person, and in twenty years, the age of the first person will be
the double of the age of the second person.
10. A group of students paid €144 for 3 tickets in the shade and 6 tickets in the sun to a bull
fight. Another group paid €66 for 2 tickets in the shade and 2 tickets in the sun. Calculate
the price of each ticket.
11. There are 420 students in a high school. Forty-two percent of “ESO” and 52% of
“Bachillerato”are girls, which means 196 in total. Calculate the amount of students in ESO
and in Bachillerato.
12. A total of 925 tickets are sold for $5925. If adult tickets cost $7,50 and children's tickets
$3,00, How many tickets of each kind were sold?
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
13. Mr. Brian has $20 000 to invest. He invests part at 6%, the rest at 7% and he earns $1 280
interest. How much did he invest at each rate?
14. There are 13 animals in the barn. Some are chickens and some are pigs. There are 40 legs in
all. How many of each animals are there?
15. Jerry was 1/3 as young as his grandfather 15 years ago. If the sum of their ages is 110, how
old is Jerry's grandfather?
16. In three more years, Miguel's grandfather will be six time as old as Miguel was last year.
When Miguel's present age is added to his grandfather's present age, the total is 68. How old
is each one now?
17. A man is 6 times as old as his son. In 4 years he will be four times as his son will be. How
old are they now?
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
Non-linear systems of equations: In this kind of systems at least one of the equation is not
linear, we usually solve these systems using the substitution method.
Example:
x− y=1
2
2
x  y =5
}
Your
Turn
1. Solve the following non-linear systems of equations:
}
a)
x · y=10
x y=7
b)
x 2 y 2=58
x 2− y 2=40
}
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
c)
1 1 1
 =
x y 20
x2y=3
d)
2x y=2
xy− x 2=0
e)
Teacher: Miguel Angel Hernández Lorenzo.
}
}
 x1 y=2x1
2x y=11
}
2. The perimeter of a rectangle is 26 cm and its surface is 40 cm 2 . Calculate its dimensions.
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
3. The sum of two real numbers is 10 and their product is 24. Calculate both numbers.
4. Find two consecutive odd numbers whose product is 49 times the square of the smaller.
5. Calculate the surface of a square knowing that the diagonal length exceeds 5 cm its side
length.
6. Calculate the perimeter and surface of a rectangle if the diagonal exceeds in 8 cm the base
and the diagonal exceeds in 16 cm the height.
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
Systems of inequalities: A system of inequalities is a collection of inequalities with the same
group of variables.
Solving a system of inequalities means finding the values of the variables that make all the
inequalities true at the same time.
Example:
x 37
x−20
}
Your
Turn
1. Solve the following systems of inequalities:
a)
2x40
x
−2x7 −3
2
}
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Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
b)
2x53
3 x−1− x30
c)
7x−1134x
2x−33x−1
d)
x83x2
x2
 x−2
2
e)
x45−2x
3x−22x1
}
}
}
}
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Teacher: Miguel Angel Hernández Lorenzo.
Unit 5: Simultaneous Equations. Mathematics 4th E.S.O.
Teacher: Miguel Angel Hernández Lorenzo.
Keywords:
equation = ecuación
linear equation = ecuación lineal
simultaneous equations (system of equations) = sistema de ecuaciones
independent system = sistema compatible determinado (única solución)
dependent system = sistema compatible indeterminado (infinitas soluciones)
inconsistent system = sistema incompatible (sin solución)
graphical method = método gráfico
numerical methods = métodos numéricos
substitution method = método de substitución
equating method = método de igualación
elimination method = método de reducción
non-linear systems of equations = sistemas de ecuaciones no lineales
systems of inequalities (inequations)= sistemas de inecuaciones
to check a solution = comprobar una solución
to verify a solution = verificar, comprobar una solución
to isolate a variable = despejar una incógnita o variable
to plug in a number for a letter = substituir una letra por un número
to solve simultaneous equations = resolver un sistema de ecuaciones
to identify = identificar
to graph = reprentar gráficamente
to set up = establecer
interval = intervalo
set solution = conjunto solución
a<b a is less than b = a es menor que b
a>b a is greater than b = a es mayor que b
a≤b a is less or equal than b = a es menor o igual que b
a≥b a is greater or equal than b = a es mayor o igual que b
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