NZQA Systems of Equations Answers
2003
1
Q1(a) x = 3, y = , z = – 2
2
Q1(b) x = 1, y = – 3, z = 5
Q5
Form the equations
4a + 2b + 4c = 100
2a + 3b + 4c = 100
31+ 3b + 2c = 100
and solve to obtain a =10, b = 20, c = 5
Q6
Form the system of equations
2h + d + 3s = 34
h + 4d + 5s = 45
3h + 2d + s = 43
and solve to obtain h =10, d = 5, s = 3
Q9
Equations are inconsistent. System represents three planes that are not parallel but do not
have a unique point of intersection (planes represent the faces of a triangular prism, or any two
planes intersect in a line that is parallel to the third plane).
2004
Q1(a)
x = -1, y = -4, z = 5
Q1(b)
a = 7, b = 2.5, c = 3
Q5(a) Form the system of equations
a = b + c +1600
b = 2c
a + b + c = 100 000
and solve to obtain a = 50 800, b = 32 800, c =16 400
Q5(b) Form the system of equations
c + b + f = 740
2.5c + 3b + 2 f = 1935
150c + 200b +120 f = 122 500
and solve to obtain c = 270, b = 320,
Q7
Two planes are parallel and the third plane intersects the other two planes.
2005
Q1
x =1, y = 3, z = 6
Q2
x = 2, y = 3, z = 8
Q5
f =150
Form the system of equations
NZQA Systems of Equations Answers
x+ y+z = 6
10x +15y + 20z = 85
2x - y - z = 0
and solve to obtain x = 2 ($10 ) , y = 3($15 ) , z =1($20 )
Q6
Form the system of equations
x + y + z = 283
x - y - z = 35
y - z = 20
and solve to obtain x =159 ( Board) , y = 72 ( Invest ) , z = 52 (Spend)
Q8
2006
Q1
Eg, Three planes which form a triangular prism, the intersection of two planes is parallel to
the third.
OR
Eg. The three planes do not all intersect and none are parallel. (Must refer to planes and not
equations)
a = 0.5, b = 5, c = -3
Q4
8a + 3b + 6c = 145
2a + 2b +10c = 130
a = 8mins, b = 7mins, c =10mins
5a + 8c = 120
Vitamin E enriched takes the least amount of time. (Must have equations, solution and decision)
2007
Q1
x = 82, y =18, z = 23
There were 82 Year 13 students involved in the school production
Q4
Form the system of equations
x + y + z = 390
5x + 8y +15z = 2520
y - 2z = 30
(where x is the number of $5 tickets, y is the number of $8 tickets and z is the number of $15
tickets)
x = 270, y = 90, z = 30
So 270 $5 tickets, 90 $8 tickets, and 30 $15 tickets were sold
Q7
Geometrical interpretation: Three planes are intersecting along a common line.
Mathematical reasons:
Eg 1: Equation (1) x 2 – equation (3), the resulting equation is: 5x - 9y + 4z = -5 which is the same
as the equation (2).
This suggests that the three planes are intersecting along a common line.
Eg 2: Equation (1) – equation (3) x 3, the resulting equation is: -17 y + 7z = -5 (4)
Equation (1) x 5 – equation (2) x 3, the resulting equation is: 17 y - 7z = 5 (5)
Adding equations (4) and (5): we get the equation 0 = 0 This means there are multiple
NZQA Systems of Equations Answers
solutions and three planes are intersecting along a common line.
Eg 3: as above with 7x + y = -3
Eg 4: as above with 17x + z = -8
2008
Q1
Q2
x = 6, y =12, z = 2 ; 2m3 od small size stones were ordered
Form the system of equations
40A + 60B +100C = 3600
A = 3+ C
B = 6 + 2C
A =15, B = 30, C =12 The cost of plant C is $12
Q3
15D + 5E + 20F = 500
D + E + F = 75
E = D + 2F
Solving produces a result that shows the system of equations is inconsistent. Therefore it is not
possible for the gardener to purchase 75 plants under the required conditions and stay within the
$500 budget.
2009
Q1(a)
x =150, y = 300, z = 50 There were 50g of macadamia nuts in the packet.
Q1(b) Form the system of equations
h + a + m = 80
3h - a + 3m = 0
8h - a = 4
(If h = weight of hazelnuts, a = weight of scorched almonds, m = weight of macadamia nuts.)
h = 8, a = 60, m =12 so 8kg of hazelnuts were ordered.
Q1(c) Since there are multiple solutions, the equations are dependent (or equivalent).
Geometrically, the three equations represent planes that meet along a common line of intersection.
Must
• explain how equations related
• give geometrical interpretation in words or a picture.
May use not independent instead of dependent.
Must refer to the planes, not equations.
2010
Q1(a)
x = 8, y =11, z = 3 There were 8 short trips, 11 medium trips and 3 long trips made.
Q1(b) Form the system of equations
x + y + z = 40
14 x + 22y + 40z = 780
z = 2y -10
(if x = number of small parcels, y = number of medium parcels, z = number of large parcels)
NZQA Systems of Equations Answers
x = 26, y = 8, z = 6 So 26 small parcels, 8 medium parcels and 6 large parcels were delivered.
2011
Q1(a) Correct = 3 marks
Incorrect = 1 mark
No attempt = –2 marks
The correct answer is worth 3 marks
Q1(b)
16c + 4d = 76
12c + 3i + 5d = 61
4c + 4i +12d = 16
Correct = 5 marks
Incorrect = 2 marks
No attempt = –1 marks
Q1(c) Equations 2 and 3 add to give equation 1, therefore there is a line of intersection between the
three planes and the system of equations has infinite solutions.