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5.11H Simultaneous Equations: Higher Tier Practice

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Solve the simultaneous equations 2x + y = 7 and x - y = 1.

(Total for Question 1 is 3 marks)

Solve the simultaneous equations 3x + 4y = 1 and 5x - 2y = 13.

(Total for Question 2 is 3 marks)

A school sells pencils and rulers. A packet of pencils and 2 rulers costs 5 pounds. Two packets of pencils and 3 rulers cost 8 pounds. Use simultaneous equations to find the cost of one packet of pencils and the cost of one ruler.

(Total for Question 3 is 4 marks)

Show that the system y = x² - x and y = 3x - 4 has exactly one real solution, and find that solution.

(Total for Question 4 is 4 marks)

Solve the simultaneous equations (1/2)x + (2/3)y = 7 and x - (3/2)y = -1. Give your answers as exact fractions or mixed numbers.

(Total for Question 5 is 4 marks)

Find the value(s) of m for which the pair of equations 2x + 3y = 5 and 4x + m y = 10
(a) has infinitely many solutions,
(b) has no solution.
Explain your answers.

(Total for Question 6 is 4 marks)

Solve the simultaneous equations y = x² - 4 and y = x + 2. Give both solutions and find the corresponding y values.

(Total for Question 7 is 4 marks)

Solve the following system of three equations in three variables:
x + y + z = 6
2x - y + 3z = 14
x + 4y - z = 2
Give the values of x, y and z.

(Total for Question 8 is 5 marks)

Priya, Kwame and Omar share 120 pounds between them. After Priya gives 10 pounds to Kwame, Priya has twice as much money as Kwame has then. After Kwame gives 5 pounds to Omar, Omar has 5 pounds less than Priya has at that time. Let P, K and O be their initial amounts. (a) Write down three simultaneous equations in P, K and O representing these facts. (b) Solve your equations to find how much each person had initially. Show all steps.

(a)Write down three equations in P, K and O to represent the situation.(3)

(b)Solve the equations to find P, K and O.(6)

(Total for Question 9 is 9 marks)

Mark scheme · 5.11H Simultaneous Equations: Higher Tier Practice

Question 1

M1 method to eliminate y, e.g. add the equations or substitute correctly
A1 x = 8/3 cao
A1 y = 5/3 cao
Answer: x = 8/3, y = 5/3

Question 2

M1 method to eliminate one variable, e.g. multiply equations to eliminate y or x
A1 x = 27/13 cao
A1 y = -17/13 cao
Answer: x = 27/13, y = -17/13

Question 3

M1 form two equations correctly, e.g. p + 2r = 5 and 2p + 3r = 8
M1 eliminate one variable correctly, e.g. multiply first equation by 2 then subtract
A1 p = 1 cao
A1 r = 2 pounds cao
Answer: packet of pencils = 1 pound, ruler = 2 pounds

Question 4

M1 substitute 3x - 4 for y to get x² - x = 3x - 4 and rearrange
dM1 complete square or factor the quadratic correctly to show (x - 2)² = 0
A1 x = 2 cao
A1 y = 2 cao
Answer: One solution: x = 2, y = 2

Question 5

M1 clear denominators correctly, e.g. multiply first by 6 and second by 2 to obtain 3x + 4y = 42 and 2x - 3y = -2
M1 eliminate one variable correctly and solve for the other
A1 x = 118/17 cao
A1 y = 90/17 cao
Answer: x = 118/17, y = 90/17

Question 6

M1 recognise that for multiples the ratios of coefficients must match, compare 4/2 and m/3 and 10/5
A1 for infinitely many solutions require 4/2 = m/3 = 10/5 leading to m = 6 cao
M1 state condition for no solutions: coefficient ratios equal but constant ratio different, and check whether this is possible with the right-hand side fixed at 10
A1 conclude no value of m gives no solutions: m = 6 makes every ratio equal (4/2 = m/3 = 10/5 = 2), giving infinitely many solutions rather than none, and any m ≠ 6 gives a unique solution cao
Answer: (a) m = 6 gives infinitely many solutions (4x + 6y = 10 is exactly twice 2x + 3y = 5). (b) No value of m gives no solutions: with the right-hand side fixed at 10, m = 6 makes the constant ratio match too (infinitely many solutions), and every other value of m gives a unique solution.

Question 7

M1 substitute x + 2 for y to get x² - 4 = x + 2 and rearrange to quadratic
M1 factorise correctly to (x - 3)(x + 2) = 0
A1 x = 3, x = -2 cao
A1 corresponding y values y = 5 and y = 0 cao
Answer: x = 3, y = 5 and x = -2, y = 0

Question 8

M1 use first equation to express z = 6 - x - y and substitute into one other equation
M1 get two linear equations in x and y, e.g. x + 4y = 4 and 2x + 5y = 8
M1 solve the 2x2 system to find x and y
A1 x = 4 cao
A1 y = 0 and z = 2 cao
Answer: x = 4, y = 0, z = 2

Question 9

(a) B1 P + K + O = 120
(a) B1 P - 10 = 2(K + 10) or equivalent rearrangement
(a) B1 P = O + 5 or O + 5 = P (from Kwame gives 5 to Omar)
(a) Answer: Equations: P + K + O = 120; P - 10 = 2(K + 10); P = O + 5
(b) M1 substitute P = O + 5 into P - 10 = 2(K + 10) and rearrange to relate O and K
(b) M1 use P + K + O = 120 with P = O + 5 to form a second equation in O and K
(b) M1 solve the two linear equations to find K and O
(b) A1 K = 13 cao
(b) A1 O = 51 cao
(b) A1 P = 56 cao
(b) Answer: Initial amounts: Priya 56 pounds, Kwame 13 pounds, Omar 51 pounds

Simultaneous Equations: Higher Tier Practice - Worksheets, Questions and Revision