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NZQA Questions
Simultaneous Equations
Report
Write equations
Solve equations and report in context
Statement
Diagram and Geometrical interpretation
Generalisation
Question 1
The gardener wants to grow three types of plants
(A, B, and C) in the new garden.
The total cost of purchasing 40 of plant A, 60 of
plant B, and 100 of plant C is $3 600.
The price of each plant A is three dollars more than
the price of each plant C.
The price of each plant B is six dollars more than
twice the price of each plant C.
Set up and solve a system of equations to find the
cost of each plant C.
Write Equations
The total cost of purchasing 40 of plant A, 60 of plant B,
and 100 of plant C is $3 600.
40A + 60B + 100C = 3600
The price of each plant A is three dollars more than the
price of each plant C.
A=C+3
The price of each plant B is six dollars more than twice
the price of each plant C.
B = 6 + 2C
Set up and solve a system of equations to find the cost of
each plant C.
Solve equations and report result in
context
40A + 60B + 100C = 3600
A=C+3
B = 6 + 2C
Using calculator
plant A costs $15 each;
plant B costs $30 each and
plant C costs $12 each
.
Statement
This system of equations represents 3 planes.
The system is independent, and consistent with
a unique solution
Diagram and geometrical
interpretation
The three planes intersect at a unique point.
Generalisation
Whenever 3 equations are independent (i.e.
none of the equations are formed from either
one or both of the other equations) and their
coefficients are not a combination of the other
equations, the system will give a unique
solution.
Question 2
The gardener plans to plant another garden using three other
types of plants (D, E, and F).
Plant D costs $15 per plant, plant E costs $5 per plant, and
plant F costs $20 per plant.
Seventy-five plants are to be bought, at a total cost of $500.
The number of plant E to be purchased is the same as the
combined total of the number of plant D and twice the
number of plant F.
Set up a system of equations for the above information and
solve them, clearly justifying your answer and carefully
explaining what your result means about the gardener’s plans.
Write equations
The gardener plans to plant another garden using three other types of
plants (D, E, and F).
Plant D costs $15 per plant, plant E costs $5 per plant, and plant F costs
$20 per plant.
15D + 5E + 20F = 500
Seventy-five plants are to be bought, at a total cost of $500.
D + E + F = 75
The number of plant E to be purchased is the same as the combined
total of the number of plant D and twice the number of plant F.
E = D + 2F
Set up a system of equations for the above information and solve
them, clearly justifying your answer and carefully explaining what your
result means about the gardener’s plans.
Solve equations and report result in
context
15D + 5E + 20F = 500 (3D + E + 4F = 100)
D + E + F = 75
D – E + 2F = 0
The calculator does not give a solution,
Show working either
Solve.
3D + E + 4F = 100
D + E + F = 75
3D + E + 4F = 100
-
D + E + F = 75
D - E + 2F = 0
+
2D+3F = 25
D + E + F = 75
D - E + 2F = 0
2D+3F = 75
2D + 3F = 25
-
2D + 3F = 75
0 = -50
Show working or
3D + E + 4F = 100
D + E + F = 75
D - E + 2F = 0
é 1 1 1 75 ù
ê
ú
ê 1 -1 2 0 ú
êë 3 1 4 100 úû
é 1 1 1 75 ù
ê
ú
ê 0 -2 1 -75 ú
êë 0 -2 1 -125 úû
Solve equations and report in context
é 1 1 1 75 ù
ê
ú
ê 0 -2 1 -75 ú
êë 0 -2 1 -125 úû
é 1 1 1 75 ù
ê
ú
ê 0 -2 1 -75 ú
êë 0 0 0 50 úû
There is no solution.
Therefore it is not possible
for the gardener to
purchase 75 plants under
the required conditions
and stay within the $500
budget.
Statement
The system of equations is independent and
inconsistent i.e. there is no solution.
Diagram and Geometrical
interpretation
The system of
equations represents
3 planes that form a
triangular prism
where the lines of
intersections of pairs
of planes are parallel
to each other.
Generalisation
Whenever combinations of multiples of the
coefficients (but not the constant terms) of two
of the equations form the third equation,
provided the planes are not parallel the system
forms a triangular prism where no solution is
possible.
You don’t need to do this.
aD + aE + aF = 75a
bD - bE + 2bF = 0
3D + E + 4F = 100
a+b = 3
a -b =1
2a = 4 Þ a = 2
b =1
Twice the first
equation plus the
second equation
gives the coefficients
of the third equation.
Question 3
Two mathematics students, Kyle and Rebecca, were
attempting to solve the following system of equations:
2x + 4y - 3z = 9
-18x -11y + 2z = -31
4x + 3y - z = 8
Kyle finds that (4,–5,–7) is a solution.
Rebecca finds that (–5,13,11) is a solution.
Explain how the equations are related, and give a
geometric interpretation of this situation.
Solve equations and report in context
2x + 4y - 3z = 9
-18x -11y + 2z = -31
4x + 3y - z = 8
Calculator does not give a solution
Show working
Solve.
2x + 4y - 3z = 9
-18x -11y + 2z = -31
4x + 3y - z = 8
+
2x + 4y - 3z = 9
- 12x + 9y - 3z = 24
10x +5y =15
-18x -11y + 2z = -31
8x + 6y - 2z = 16
10x +5y =15
10x + 5y = 15
-
10x + 5y = 15
0=0
Solve equations and report in context
There are an infinite number of solutions, so we
write the solutions in general form.
10x +5y = 15
2x + y = 3 Þ y = 3- 2x
4x + 3y - z = 8
Þ z = 4x + 3y - 8 = 4x + 3( 3- 2x ) - 8 = -2x +1
1- z
Þx=
2
Solve equations and report in context
There are an infinite number of solutions, so we
write the solutions in general form.
2x + y = 3Þ 2x = 3- y
4x + 3y - z = 8
Þ z = 4x + 3y - 8 = 2 ( 3- y) + 3y - 8 = y - 2
1- z
Þ y = z + 2, x =
2
æ 1- z
ö
Solution is ç
, z + 2, z ÷
è 2
ø
Solve equations and report in context
Kyle finds that (4,–5,–7) is a solution.
Rebecca finds that (–5,13,11) is a solution.
æ 1- z
ö
Solution is ç
, z + 2, z ÷
è 2
ø
z = -7 Þ x = 4, y = -5 Kyle's solution
z = 11 Þ x = -5, y = 13 Rebecca's solution
Statement
The system of equations represents 3 planes.
The system of equations is dependent and
consistent with an infinite number of solutions.
Diagram and Geometrical
interpretation
Generalisation
Where one of the equations is formed from a
combination of multiples of the other two, there
will be an infinite number of solutions because
they intersect on a line.
2ax + 4ay - 3az = 9a
-18bx -11by + 2bz = -31b
4x + 3y - z = 8
2a -18b = 4
4a -11b = 3
a = 0.2, b = -0.2
1/5 of the first
equation, then
subtract 1/5 of the
second equation.