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Simultaneous
Equations
Example
x +y=8
x - y= 4
2x = 12
x= 6
x +y = 8
6+y = 8
y = 2
Solve the equations
a) 4x + y = 9
x−y=1
e) 4x + 3y = 27
5x − 2y = 28
b) 2x − 5y = −6
−x + 3y = 4
f) −3x + 2y = −8
4x − 9y = −2
c) −3x + 4y = 4
6x − 5y = 4
g) 2x − 3y = 5
5x + 2y = −16
d) 4x − 2y = −10
−3x + 8y = 14
h) 2x + 5y = −3
9x + 4y = −13
Solve the equations
a) x − y = −5
x + 3y = 27
e) 2x − 3y = −8
5x + y = 14
b) 2x + 3y = 10
−3x + 2y = −41
f) 3x + y = −2
−2x − 3y = 13
c) 3x + 2y = 16
4x + y = 13
g) 2x + y = 12
3x − 2y = 13
d) 4x − 3y = 5
9x − 2y = 16
h) −3x − 5y = −8
11x − 2y = −2
Problem Solving with
Simultaneous Equations
• For the cinema
• 2 adults’ tickets and 5 children’s tickets cost £26
• 4 adults’ tickets and 2 children tickets cost £28
• Find the cost of each kind of ticket.
• Read the problem
• Introduce two letters
• Write two equations that describe the information
• Solve the problem using simultaneous equations
PROBLEMS WITH
SIMULTANEOUS EQUATIONS
Question 1
a) The tickets for a sports club cost £2 for members and £3 for
non- members. The total ticket money collected was £580 x
tickets were sold to members and y tickets were sold to
non-members. Use this information to write down an
equation involving x and y
b) 250 people bought raffle tickets for the disco. Write down
another equation involving x and y.
c) How many tickets were sold to members ?
Question 2
Alloys are made by mixing metals. Two different alloys are made using
iron and lead. To make the first alloy, 3cm3 of iron and 4 cm3 of lead
are used. This alloy weighs 65 grams
a)
Let x grams be the weight of 1 cm3 of iron and y grams be the
weight of 1 cm3 of lead. Write down an equation in x and y
which satisfies the above equation.
b) To make the second alloy, 5 cm3 of iron and 7cm3 of lead are
used . This alloy weighs 112 grams. Write down a second
equation in x and y which satisfies this condition.
c)
Find the weight of 1 cm3 of iron and the weight of 1cm3 of lead
Question 3
• A rectangular window has length , L cm and breadth, B cm . A
security grid is made to fit this window. The grid as 5
horizontal wires and 8 vertical wires
a) The perimeter of the window is 260 cm. Use this
information to write down an equation involving L and B .
b) In total, 770 cm of wire is used. Write down another
equation involving L and B
c) Find the length (L) and breadth (B) of the window
Question 4
• The senior classes at High School A and High School B planned
separate trips to New York City. The senior class at High School
A rented and filled 1 van and 6 buses with 372 students. High
School B rented and filled 4 vans and 12 buses with 780
students. Each van and each bus carried the same number of
students. How many students can a van carry? How many
students can a bus carry?
Additional Questions
1)
The difference of two numbers is 3, and the sum of three times the larger one
and twice the smaller one is 19. Find the two numbers.
2)
The sum of four times the first number and three times the second number is
15. The difference of three times the first number and twice the second
number is 7. Find the numbers.
3)
9 pens and five pencils cost $3.2, and 7pens and 8 pencils cost $2.9. Find the
unit price for each pen and pencil.
4)
Two runners start from the same point at the same time. They will be 4 miles
apart at the end of two hours if running in the same direction, and they will
be 16 miles apart at the end of one hour if running in opposite directions. Find
their speeds.
Additional Questions cont.
5)
A solution containing 12% alcohol is to be mixed with a solution containing 4%
alcohol to make 20 gallons of solution containing 9% alcohol. How much of
each solution should be used?
6)
If twice the son’s age in years is added to the father’s age, the sum is 70. But if
the twice of father’s age is added to the son’s age, the sum is 95. Find the ages
of father and son.
7)
2 tables and 3 chairs together cost 2000 dollars whereas 3 tables and 2 chairs
together cost 2500 dollars. Find the cost of a table and a chair.
8)
3 bags and 4 pens together cost 257 dollars whereas 4 bags and 3 pens
together cost 324 dollars. Find the cost of a bag and 10 pens