(a)Complete the table of values for y = x + 1. x: -2, -1, 0, 1, 2, 3 y: __, __, 1, __, __, __(2)
(b)On the grid, draw the graph of y = x + 1 for values of x from -2 to 3.(2)
(Total for Question 1 is 4 marks)
2
The lines y = 3x - 2 and y = 1 are already drawn on the grid.
(Total for Question 2 is 1 mark)
3
The diagram shows two straight lines, l1 and l2, drawn on the same grid. l1 has equation y = 2x - 1 and l2 has equation y = -x + 5.
(a)Write down the coordinates of the point where l1 and l2 intersect.(1)
(b)Hence write down the solution to the simultaneous equations y = 2x - 1 and y = -x + 5.(2)
(Total for Question 3 is 3 marks)
4
The graphs of two straight lines intersect at the point (-2, 5). Which of the following is the solution to the pair of simultaneous equations represented by the lines?
A) x = 5, y = -2
B) x = -2, y = 5
C) x = 2, y = -5
D) x = -5, y = 2
(Total for Question 4 is 1 mark)
5
Show that the point (2, 3) is the solution to the simultaneous equations y = x + 1 and y = -2x + 7.
(Total for Question 5 is 2 marks)
6
By drawing suitable straight lines on the grid, solve the simultaneous equations y = x + 3 and y = 2x + 1.
(Total for Question 6 is 4 marks)
7
(a) Rearrange 2x + y = 8 to make y the subject. (b) The line y = x - 1 is already drawn on the grid. By completing a table of values and drawing the graph of y = 8 - 2x on the same grid, solve the simultaneous equations 2x + y = 8 and y = x - 1.
(a)Rearrange 2x + y = 8 to make y the subject.(1)
(b)By completing a table of values and drawing the graph of y = 8 - 2x on the same grid, solve the simultaneous equations 2x + y = 8 and y = x - 1.(4)
(Total for Question 7 is 5 marks)
8
The line y = 4 - x is already drawn on the grid. By drawing the graph of y = 2x - 5 on the same grid, solve the simultaneous equations y = 4 - x and y = 2x - 5.
(Total for Question 8 is 4 marks)
9
Two straight lines have equations y = 2x + 3 and y = 2x - 1.
(a)Without drawing the graphs, state whether the two lines are parallel. Give a reason for your answer.(2)
(b)Hence explain what this tells you about the solution to the simultaneous equations y = 2x + 3 and y = 2x - 1.(1)
(Total for Question 9 is 3 marks)
10
A cinema sells adult tickets for x pounds and child tickets for y pounds. The graphs of 2x + y = 23 and x + 3y = 29 are drawn on the grid, where the axes show ticket price in pounds.
(a)Use the graph to find the price of an adult ticket and the price of a child ticket.(2)
(b)Find the total cost for a family of 2 adults and 2 children.(2)
(Total for Question 10 is 4 marks)
11
Priya is comparing two mobile phone tariffs. Tariff A: monthly cost C (pounds) = 20 + 0.02m, where m is the number of minutes used. Tariff B: monthly cost C (pounds) = 8 + 0.06m. The graphs of Tariff A and Tariff B are shown on the grid for 0 ≤ m ≤ 500.
(a)Use the graph to write down the number of minutes for which the two tariffs cost the same amount.(1)
(b)A customer uses 150 minutes in a month. State which tariff is cheaper, and by how much.(3)
(c)Bilal uses more than 400 minutes a month. Which tariff should he choose? Give a reason using the graph.(2)
(Total for Question 11 is 6 marks)
12
Two straight lines are drawn on the same grid and cross at a single point.
(Total for Question 12 is 2 marks)
13
The diagram shows two straight lines drawn on a grid. Line 1 passes through (0, 6) and (4, -2). Line 2 passes through (0, -3) and (3, 3).
(a)Find the equation of Line 1 in the form y = mx + c.(2)
(b)Find the equation of Line 2 in the form y = mx + c.(2)
(c)Hence find the coordinates of the point of intersection of the two lines, showing your method.(3)
(Total for Question 13 is 7 marks)
14
(a) Rearrange 3x - 2y = 6 to make y the subject. (b) Rearrange x + y = 7 to make y the subject. (c) By drawing suitable graphs on the grid, solve the simultaneous equations 3x - 2y = 6 and x + y = 7.
(a)Rearrange 3x - 2y = 6 to make y the subject.(1)
(b)Rearrange x + y = 7 to make y the subject.(1)
(c)By drawing suitable graphs on the grid, solve the simultaneous equations 3x - 2y = 6 and x + y = 7.(4)
(Total for Question 14 is 6 marks)
15
The graphs of y = x2 - 1 and y = 3 are drawn on the same grid. How many points of intersection are there, and what does this tell you about the number of solutions to the equation x2 - 1 = 3?
A) 0 intersections, no solutions
B) 1 intersection, 1 solution
C) 2 intersections, 2 solutions
D) 3 intersections, 3 solutions
(Total for Question 15 is 1 mark)
16
The curve y = x2 is drawn on the grid.
(a)On the same grid, draw the graph of y = x + 2.(2)
(b)Hence write down the two pairs of values (x, y) that solve the simultaneous equations y = x2 and y = x + 2.(2)
(Total for Question 16 is 4 marks)
Mark scheme · 5.12 Solving Simultaneous Equations Graphically
Question 1
(a) B1 at least 3 of the 5 missing values correct
(a) B1 all 5 missing values correct (cao)
(a) Answer: x = -2, -1, 0, 1, 2, 3 gives y = -1, 0, 1, 2, 3, 4
(b) M1 at least 4 points plotted correctly, ft from table in (a)
(b) A1 correct ruled straight line drawn from x = -2 to x = 3
(b) Answer: Straight line through (-2,-1), (-1,0), (0,1), (1,2), (2,3), (3,4)
Question 2
B1 x = 1, y = 1 (oe)
Answer: x = 1, y = 1
Question 3
(a) B1 (2, 3)
(a) Answer: (2, 3)
(b) B1 x = 2 (ft from (a))
(b) B1 y = 3 (ft from (a))
(b) Answer: x = 2, y = 3
Question 4
B1 B
Answer: B) x = -2, y = 5
Question 5
M1 substitutes x = 2 into both equations
A1 shows y = 3 from both equations, cso, so (2,3) satisfies both equations
Answer: y = 2 + 1 = 3 and y = -2(2) + 7 = 3, both give y = 3, so (2,3) satisfies both equations
Question 6
M1 draws y = x + 3 correctly through at least 3 correct points, e.g. (0,3), (1,4), (-2,1)
M1 draws y = 2x + 1 correctly through at least 3 correct points, e.g. (0,1), (1,3), (2,5)
A1 identifies intersection point at (2,5)
A1 states solution x = 2, y = 5 (ft from their graph)
Answer: x = 2, y = 5
Question 7
(a) B1 y = 8 - 2x (oe)
(a) Answer: y = 8 - 2x
(b) M1 table of at least 3 correct values for y = 8 - 2x, e.g. x=0,y=8; x=2,y=4; x=4,y=0
(b) M1 correct line y = 8 - 2x drawn, ft from table
(b) A1 identifies intersection at (3,2)
(b) A1 states solution x = 3, y = 2 (ft)
(b) Answer: x = 3, y = 2
Question 8
M1 at least 3 correct points for y = 2x - 5, e.g. (0,-5), (2,-1), (3,1)
A1 correct line y = 2x - 5 drawn
B1 identifies intersection at (3,1)
B1 states solution x = 3, y = 1 (ft)
Answer: x = 3, y = 1
Question 9
(a) B1 yes, the lines are parallel
(a) B1 reason: both lines have the same gradient (coefficient of x = 2)
(a) Answer: Yes, the lines are parallel because they both have gradient 2
(b) B1 the equations have no solution because parallel lines with different y-intercepts never meet
(b) Answer: There is no solution, since the lines never intersect
Question 10
(a) B1 adult ticket = £8 (x = 8)
(a) B1 child ticket = £7 (y = 7)
(a) Answer: Adult ticket = £8, child ticket = £7
(b) M1 2(8) + 2(7)
(b) A1 £30 cao
(b) Answer: £30
Question 11
(a) B1 300 minutes (awrt 300, accept range 290 to 310)
(a) Answer: 300 minutes
(b) M1 substitutes m = 150 into both C = 20 + 0.02m and C = 8 + 0.06m
(b) A1 Tariff A = £23, Tariff B = £17
(b) A1 Tariff B is cheaper by £6
(b) Answer: Tariff B is cheaper, by £6
(c) B1 Tariff A
(c) B1 reason: for m > 300, the graph of Tariff A lies below the graph of Tariff B, so Tariff A costs less
(c) Answer: Tariff A, because for minutes above 300 its line is below Tariff B's line
Question 12
B1 the point of intersection is the only point that lies on both lines
B1 its coordinates satisfy both equations at the same time, so they give the solution to the pair of simultaneous equations
Answer: The intersection point is the only point common to both lines, so its (x,y) coordinates are the solution that satisfies both equations simultaneously
Question 13
(a) M1 gradient = (-2 - 6) / (4 - 0) = -2
(a) A1 y = -2x + 6 (oe)
(a) Answer: y = -2x + 6
(b) M1 gradient = (3 - (-3)) / (3 - 0) = 2
(b) A1 y = 2x - 3 (oe)
(b) Answer: y = 2x - 3
(c) M1 sets -2x + 6 = 2x - 3
(c) M1 correctly solves for x: 9 = 4x, x = 2.25
(c) A1 y = 1.5, coordinates (2.25, 1.5) cao
(c) Answer: (2.25, 1.5)
Question 14
(a) B1 y = 1.5x - 3 (oe, e.g. y = (3x - 6) / 2)
(a) Answer: y = 1.5x - 3
(b) B1 y = 7 - x
(b) Answer: y = 7 - x
(c) M1 draws y = 1.5x - 3 correctly, ft from (a), e.g. through (0,-3), (2,0), (4,3)
(c) M1 draws y = 7 - x correctly, ft from (b), e.g. through (0,7), (4,3), (7,0)
(c) A1 identifies intersection at (4,3)
(c) A1 states solution x = 4, y = 3 (ft)
(c) Answer: x = 4, y = 3
Question 15
B1 C
Answer: C) 2 intersections, 2 solutions
Question 16
(a) M1 at least 2 correct points, e.g. (0,2), (2,4)