1.	x = 6
y = –2	4
	Eg eqn (1) ´ 2 then add to eqn (2) ´ 3 ® 13x = 78
x = 6
Eg 2 ´ “6” + 3y = 6
y = –2
M1 correct process to eliminate either x or y (condone one error)
A1 cao for non-eliminated one
M1 (dep on 1st M1) for correct substn of their found value
A1 cao (need both)
[4]

2.	x = 4, y = - 1	4
	4x - 6y = 22
15x + 6y = 54
19x = 76
M1 for coefficients of x or y the same followed by correct operation, one arithmetical error
A1 cao
M1(dep) for correct sub for other variable
A1 cao
Trial and improvement 0 marks unless both x and y correct values found
[4]

3.	x2 + 6x + 9
x = 2 and y = 5
or
x = – 5 and y = – 2	7
	(x + 3)2 + x2 = 29
x2 + 6x + 9 + x2 = 29
2x2 + 6x + 9 = 29
2x2 + 6x – 20 = 0
2(x – 2)(x + 5) = 0
M1 for rearranging to y = x + 3
M1 for correct substitution for their y to give (ax + b)2
+ x2 = (29)
A1 (Ö) for correct exp of (ax + b)2 seen
M1 for reduction to a 3 term quadratic = 0
M1 (dep on 3rd M1) for correct factorisation (Ö ) or correct use of formula (Ö ) or completing the square (Ö )
A1 for at least 2 correct values of x or y
A1 cao, values must be paired

4.	x = 4 ½
y = –3	4
	18x – 6y = 99	12x – 4y = 66
8x + 6y =18	12x + 9y = 27
26x = 117	– 13y = 39
M1 correct process to eliminate either x or y (condone one error)
A1 cao for non – eliminated one.
M1 (dep on 1st M1) for correct substitution of their found value.
A1 cao (need both)
[4]

5.	(a)	x2 + 62 = 25
x2 = – 11	2
M1 substitutes y = 6 to get x2 + 36 = 25
A1 deduces x2 = 25 – 36 < 0; impossible, (so line does not intersect curve)
(b)	x = 3
y = 4
x = – 1.4
y = –4.8	6
	y2 = (2x – 2)2
x2 + (2x – 2)2 = 25
x2 + 4x2 – 8x + 4 = 25
5x2 – 8x + 4 – 25 = 0
5x2 – 8x – 21 = 0
(5x + 7)(x – 3) = 0 or

So x = 3, or x = –1.4
Subs into either equation gives:
y = 4, or y = – 4.8
M1 y2=(2x – 2)2 seen or implied
A1 x2 + 4x2 – 8x + 4 = 25 (oe expanded form)
M1 correct attempt to solve 3 term quadratic equation
A1 x = 3, x = –1.4
M1 (dep on previous Ms) Sub one value of x into
either equation gives:
A1 y = 4, y = –4.8
NB: incorrectly matched pairs loses the final A1.
[8]

6.	(a)	Circle centre O Line	3
B1 correct circle, within overlay
B2 correct line tol ± 1mm at (4, 4) and (0, –2)
(B1 for any straight line with the correct intercept on the y axis)
(b)	x = 6.4,
y = 7.7
x = – 4.6,
y = – 8.9	2
B2 Two paired solutions, ft from a line and a curve with at least B1 scored in (a)
B1 Any two correct values, , ft from a line and a curve with at least B1 scored in (a)
Tol ± 0.2
(c)	q = 9	2
	(x + 3)2 – 9
B1 for x2 + 6x + 9 seen
B1 for q = 9
(d)	3,10	3
	(x + 3)2 – 9 + (y – 2)2 – 4 – 87 = 0
(x + 3)2 + (y – 2)2 = 100
M1 for completing the square
A1 for (y – 2)2 – 4 seen
A1 any correct answer
[10]

7.	x = 3
y = 0.5	3
M1 for coefficients of x or y the same followed by correct operation, condone one arithmetical error
M1 (dep) for substituting found value in one equation
A1 cao
SC: B1 for one correct answer only if Ms not awarded
[3]

8.	Straight line through (0,1) and (1,3)
x = 1.8
y = 4.7;
x = –2.6
y = –4.2	3
M1 draw y = 2x + 1 or any st line with positive gradient thro’ (0,1) or any line with gradient 2. Must cut circle at least once.
A1 first solution (±1 square, ft but dep on a single straight line)
A1 cao second solution (±1 square)
[3]

9.	(a)	(0,2)
(1,0)
(2,0)	3
B1 for graph translated through or 
B1 for y intercept at (0, 2) ± 2 mm
B1 for x intercept at (1, 0) and (2, 0) ± 2 mm
(b)	B1 correct sketch showing stretch SF3 parallel to the y-axis	1
(c)	3(x2–x) = 4–4x
3x2 + x – 4 = 0
(3x + 4)(x – 1) = 0
x–1 or x = – 
Subs to get y=0 or y= 9 
(1,0)
	5
M1 for equating the RHS of each equation
M1 for reduction to 3 term quadratic( = 0)
M1 for (3x ± a)(x ± b) (=0) with ab = – 4
or , allowing sign errors in b and c.
A1 for (1, 0)
A1 for oe
SC B1 for (1, 0) no marks awarded previously
SC 4/5 if the correct algebra is used to find the two correct x
values, but the two y values are not found or are incorrect
[9]

10.	eg eqn(1) ×4 then subtract eqn(2) ×3	13y = 65
x = –3
or eqn(2) ×7 then subtract eqn(1) ×5	13x = –39
eg 4x + 5 × ‘5’ = 13
y = 5		4
M1 correct full process to eliminate either x or y (condone one error)
A1 cao either y = 5 or for x = –3
M1 (dep. on 1st M1) for correct substitution of their found value into one of the eqn’s
A1 cao (both needed)
[4]

11.	a = 3
b = –2	3
M1 for a complete method which leads to a single equation in a or b only (allow 1 error)
M1 (dep) substitute found value of a or b into one equation
A1 cao
SC B1 for one correct answer only if Ms not awarded,
a = 3 or b = –2
[3]

12.	x = 3, y = -2	2
B1
B1 cao
[2]

13.	x = 3
y = -0.5	4
	9x - 12y = 33
10x + 12y = 24
	19x = 57
x = 3
	15 + 6y = 12
y = -0.5
M1 for correct process to eliminate either x or y (condone 1 error)
A1 cao for non-eliminated variable
M1 (dep on 1st M1) for correct substitution of their found value
A1 cao (need both)
[4]

14.	x = -2 or 6
y = 6 or -2	7
	x = 4 – y
(4 – y)2 + y2 = 40
2y2 – 8y – 24 = 0
(y – 6)(y + 2)
B1 for y = 4– x or x = 4 – y
M1 for subs of “4 - x” or “4 - y” into x2 + y2 = 40
M1 for expansion of (4 - y)2 or (4 - x)2, 3 out of 4 terms correct
A1 for 2y2 – 8y – 24 = 0 or 2x2 – 8x – 24 = 0
M1 for correct factorisation or correct substitution into formula [ft quadratic equation of type ax2 + bx + c = 0, a, b, c ¹ 0]
A1 for both x values
A1 for both y values
[7]

15.	(5.5, 14)	3
	4x – 8 = 2x + 3
2x = 11
x = 5.5
	y = 2 × 5.5 + 3
M1 for 4x – 8 = 2x + 3 or correct method to eliminate x or y
A1 for x = 5.5 oe
A1 for y = 14
(SC: If no marks awarded B1 for either x = 5.5or y = 14)
[3]

16.	y = –2, x = –3
y = 1.2 oe, x = 3.4 oe	7
	x = 1 + 2y
(1 + 2y)2 + y2 + y2 = 13
1 + 4y + 4y2 + y2 = 13
5y2 +4y – 12 = 0
(5y – 6)(y + 2) = 0
B1 for x = 1 + 2y
M1 for subs. x = 1 + 2yor y =  into x2 + y2 = 13
M1 for expansion of (1 + 2y)2, 3 out of 4 terms correct
A1 for 5y2 +4y – 12 = 0 or 5x2 – 2x – 51 = 0
M1 for correct method to solve a 3 term quadratic equation
A1 for y = –2 and y = 1.2
A1 for x = –3 and x = 3.4
(SC: If no marks awarded
B1 for x = –3 and y = –2
or x = 3.4 and y = 1.2)

17.	y = 3 + 2x
x2 + (3 + 2x )2 = 18
x2 + 9 + 12x + 4x2 – 18 = 0
5x2 + 12x – 9 = 0
(5x – 3)(x + 3) = 0
x = –3         y = –3
x =  oe y = 4oe	7
M1 for y = 3 + 2x or x = (y–3) ÷ 2
M1 for “correct” substitution
M1 for “correct” expansion, 3 out of 4 terms correct
A1 for 5x2 + 12x – 9 = 0 or 5y2 – 6y – 63 = 0
M1 for correct attempt to solve a 3 term quadratic equation
A1 for both x (or y) values
A1 for both correct solutions
[SC: If no marks earnt then B1 for x = –3 and y = –3
OR x =  and y = ]
[7]