Question 1
LEVEL 8
Solve the following simultaneous equations:
x^2 + y^2 = 100
x - y = 2
Select the correct answer from the list below:
A: x = 8, y = 6 or x = -6, y = -8
B: x = 4, y = 2 or x = -3 y = -8
C: x = 8, y = 6 or x = -6, y = 6
D: x = 4, y = 2 or x = -3, y = 6
CORRECT ANSWER: A: x = 8, y = 6 or x = -6, y = -8
WORKED SOLUTION:
Subsitue y = x-2 into the first equation
x^2 +(x-2)^2=100
We can expand and simplify this equation to find x
x^2 +x^2-4x+4 = 100
2x^2 -4x-96 = 0
x^2 -2x-48 = 0
(x-8)(x+6) = 0
x=8 or x=-6
Subsituting this value back into one of the original equations to find y
y = x-2
y=6 or y=-8
ANSWER: x = 8, y = 6 or x = -6, y = -8
Question 2
LEVEL 8
Solve the following simultaneous equations:
y = x^2 - 3x + 4
y - x = 1
Select the correct answer from the list below:
A: x = 4, y = 3 or x = -1, y = 2
B: x = 2, y = 3 or x = 2, y = 2
C: x = 3, y = 4 or x = 1, y = 2
D: x = 1, y = 1 or x = 1, y = 2
CORRECT ANSWER: C: x = 3, y = 4 or x = 1, y = 2
WORKED SOLUTION:
Substitute y = 1+x into the first equation
1+x = x^2 - 3x + 4
We can simplify this equation to find x
x^2 -4x +3 = 0
(x-3)(x-1) = 0
x=3 or x=1
Substituting this value back into one of the original equations to find y
y = 1+x
y=4 or y=2
ANSWER: x = 3, y = 4 or x = 1, y = 2
Question 3
LEVEL 8
Solve the following simultaneous equations:
y = x^2 + 7x - 2
y = 2x - 8
Select the correct answer from the list below:
A: x = -2, y = -8 or x = -1, y = -2
B: x = -4, y = -1 or x = -2, y = -8
C: x = -3, y = -14 or x = -2, y = -12
D: x = -2, y = -14 or x = -3, y = -12
CORRECT ANSWER: B: x = -3, y = -14 or x = -2, y = -12
WORKED SOLUTION:
Substitute y = 2x - 8 into the first equation
2x - 8 = x^2 + 7x - 2
We can simplify this equation to find x
x^2 + 5x +6 = 0
(x+3)(x+2) = 0
x=-3 or x=-2
Substituting this value back into one of the original equations to find y
y = 2x - 8
y=-14 or y=-12
ANSWER: x = -3, y = -14 or x = -2, y = -12
Question 4
LEVEL 8
Solve the following simultaneous equations:
x^2 + y^2 = 8
y = x + 4
Select the correct answer from the list below:
A: x = -2, y = 2
B: x = 2, y = -2
C: x = 2, y = 6
D: x = -2, y = 2, or x = 2, y = -2
CORRECT ANSWER: A: x = -2, y = 2
WORKED SOLUTION:
Substitute y=x+4 into the first equation
x^2 + (x+4)^2 = 8
We can expand and simplify this equation to find y
x^2 + x^2+8x+16 = 8
2x^2+8x+8=0
x^2+4x+4=0
(x+2)^2=0
x=-2
Substituting this value back into one of the original equations to find y
y= x+4
y=-2+4
y=2
ANSWER: x = -2, y = 2
Question 5
LEVEL 8
Solve the following simultaneous equations:
y = x^2
y = x + 2
Select the correct answer from the list below:
A: x = 2, y = 1 or x = 2, y = 6
B:x = -1, y = 3 or x =-1, y = 4
C: x = -4, y = 1 or x = 6, y = 2
D: x = -1, y = 1 or x = 2, y = 4
CORRECT ANSWER: D: x = -1, y = 1 and x = 2, y = 4
WORKED SOLUTION:
Substitute y=x+2 into the first equation
x+2 =x^2
We can simplify this equation to find x
x^2-x-2=0
(x+1)(x-2)=0
x=-1 or x=2
Substituting these values back into one of the original equations to find y
y=x+2
y=1 or y=4
ANSWER: x = -1, y = 1 and x = 2, y = 4
Question 6
LEVEL 8
Solve the following simultaneous equations:
x^2 + y^2 = 5
x - 2y = 5
Select the correct answer from the list below:
A: x = 2, y = -1
B: x =9, y = -2
C: x = -2, y = 1
D: x = 1, y = -2
CORRECT ANSWER: D: x = 1, y = -2
WORKED SOLUTION:
Substitute x=2y+5 into the first equation
(2y+5)^2 + y^2 = 5
We can expand and simplify this equation to find y
(2y+5)^2 + y^2 = 5
(4y^2+20y+25) + y^2 = 5
5y^2+20y+20 = 0
y^2+4y+4 = 0
(y+2)^2 = 0
y=-2
Substituting this value back into one of the original equations to find x
x - 2(-2) = 5
x+ 4 = 5
x=1
ANSWER: x = 1, y = -2