Question 1
1(a):
Solve the simultaneous equations below.
6x+3y=12\\ 2x+6y=14ANSWER: Multiple Answers (Type 1)
Answer: x=1, y=2
Workings:
Multiply 2nd equation by 3 to get the same quantities of x.
6x+3y=12\\
6x+18y=42
Subtracting the first equation from the second gives
15y=30
y=2
Substitute this value for y into equation 1 to find x
6x+3\times 2 =12
6x=6
x=1
Marks = 3
1(b):
Solve the simultaneous equations below.
2x-5y=16
ANSWER: Multiple Answers (Type 1)
Answer: x=3, y=-2
Workings:
Multiply the first equation by 3 and the second by 2 to get the same quantities for x.
6x-15y=48
6x+4y=10
Subtract the first equation from the second.
19y=-38
y=-2
Substitute this value for y into equation one.
2x-5y=16
2x-5\times -2=16
x=3
Marks = 3
Question 2
2(a):
Solve the simultaneous equations below.
2x+4y=14\\ 4x-4y=4ANSWER: Multiple Answers (Type 1)
Answer: x=3, y=2
Workings:
Add the two equations together
6x=18\\ x=3Substitute into Equation One
2\times 3 + 4y=14\\ y=2Marks = 3
2(b):
Solve the simultaneous equations below.
3x-y=23\\ 2x+3y=8ANSWER: Multiple Answers (Type 1)
Answer: x=7, y=-2
Workings:
Multiply Equation One by 2 and Equation Two by 3 to get the same quantities of x.
6x-2y=46
6x+9y=24
Subtract Equation One from Equation Two
-11y=22
y=-2
Substitute into Equation One
6x-2(-2 )=46
x=7
Marks = 3
Question 3
Two different families pay for entry into a water park.
Family 1 has 2 adults and 3 children and costs a total of £20 to enter the park.
Family 2 has 1 adult and 4 children and costs a total of £15 to enter the park.
Work out the cost of the adult ticket, and the child ticket in £.
ANSWER: Multiple Answers (Type 1)
Answer: c=2 & a=7
Workings:
Create two equations to represent each family.
2a+3c=20
a+4c=15
Multiply equation two by 2 so each equation has the same a quantities.
2a+3c=20
2a+8c=30
Subtract equation one from equation two
5c=10
c=2
Substitute value into equation one
2a+3\times 2=20
a=7
Marks = 4
Question 4
Sophie is selling student and parent tickets for a school performance.
On night one, she sells 50 student tickets and 80 parent tickets and makes £340.
On night two, she sells 25 student tickets and 50 parent tickets and makes £200.
What is the cost for 1 student ticket and the cost for 1 parent ticket in £?
ANSWER: Multiple Answers (Type 1)
Answer: s=2, p=3
Workings:
Set up two equations, one for each night.
50s+80p=340
25s+50p=200
Multiply equation two by 2 to get equal quantities for s.
50s+80p=340
50s+100p=400
Subtract equation one from equation two
20p=60
p=3
Substitute into equation one
50s+80\times 3 =340
s=2
Marks = 4
Question 5
5(a)
Solve the simultaneous equations below.
4x+8y=-4\\ 2y -5x=23ANSWER: Multiple Answers (Type 1)
Answer: x=-4, y=1.5
Workings:
Multiply equation two by 4 to get the same quantities of y for each equation
4x+8y=-4\\
8y-20x=92
Subtract equation two from equation one
24x=-96
x=-4
Substitute into equation one
4\times -4 +8y=-4
y=1.5
Marks = 3
5(b)
Give the coordinates for the point of intersection for the lines below.
4x+8y=-4\\ 2y -5x=23ANSWER: Multiple Choice (Type 1)
A: (2, 6.5)
B: (2.5, 4)
C: (-4, 1.5)
D: (1.5, 3.5)
Answer: C
Workings:
The point of intersection is where both lines have the same value for x and y in the format (x,y).
In this case this will be (-4, 1.5)
Marks = 1
Question 6
Andrew goes to the shop to buy some apples and bananas.
He goes to purchase 5 apples and 4 bananas, and the total comes to £5.70.
Unfortunately, he doesn’t have enough money, so he puts back 1 apple and 2 bananas.
The new total is £3.60. What is the cost of 1 apple and the cost of 1 banana in £?
ANSWER: Multiple Answers (Type 1)
Answer: a=0.50, b=0.80
Workings:
Create two equations to represent the original transaction and the final one.
5a+4b=570
4a+2b=360
Multiply equation two by 2 to get equal quantities for b.
5a+4b=570
8a+4b=720
Subtract equation one from equation two
3a=150
a=50=£0.50
Substitute into equation one
5\times 50+4b=570
b=80=£0.80
Marks = 3
Question 7
Two simultaneous equations are given below, where p and q are constants.
3x-py=4\\ 4x-3y+q=0The solution to these equations is x=1, y=2.
Find the value of p and q.
ANSWER: Multiple Answer (Type 1)
Answer: p=\dfrac{-1}{2}, q=2
Workings:
Substitute in the values for x and y.
3-2p=4
q-2=0
q=2
p=\dfrac{-1}{2}
Marks = 4
Question 8
Examine the rectangle below:
8(a)
The area of the rectangle is 88cm^2.
What is the value of a in cm?
ANSWER: Simple Text Answer
Answer: 11
Workings:
The area is equal to 8a
8a=88
a=11
Marks = 1
8(b)
Using the information from the diagram, what is the value of x and y?
ANSWER: Multiple Answers (Type 1)
Answer: x=2.5, y=6
Workings:
2x+y=11
-4x+3y=8
Multiply equation one by -2 to get equal quantities for x in each equation.
-4x-2y=-22
-4x+3y=8
Subtract equation one from equation two
5y=30
y=6
Substitute into equation one
2x+6=11
x=2.5
Marks = 3