Complete the table of values for y = 2x - 1. x: -2, -1, 0, 1, 2, 3 y: __, __, __, 1, __, __
(Total for Question 1 is 2 marks)
2
The graphs of y = 2x - 6 and y = -x + 6 are drawn on the grid. Write down the solution of the simultaneous equations y = 2x - 6 and y = -x + 6.
(Total for Question 2 is 2 marks)
3
The graphs of y = x - 1 and y = -2x - 7 are drawn on the grid. Write down the solution of the simultaneous equations y = x - 1 and y = -2x - 7.
(Total for Question 3 is 2 marks)
4
The graphs of x = 3 and y = 2x - 4 are drawn on the grid. Write down the solution of the simultaneous equations x = 3 and y = 2x - 4.
(Total for Question 4 is 2 marks)
5
The graphs of y = 1 and y = 3x - 5 are drawn on the grid. Write down the solution of the simultaneous equations y = 1 and y = 3x - 5.
(Total for Question 5 is 2 marks)
6
The graphs of two straight lines intersect at the point (-3, 4). Which of the following is the solution to the pair of simultaneous equations represented by the lines?
A) x = 4, y = -3
B) x = -3, y = 4
C) x = 3, y = -4
D) x = -4, y = -3
(Total for Question 6 is 1 mark)
7
Rearrange 3x + y = 9 to make y the subject, ready for plotting.
(Total for Question 7 is 1 mark)
8
Show that the point (3, -1) is the solution to the simultaneous equations y = 2x - 7 and y = -3x + 8.
(Total for Question 8 is 2 marks)
9
The lines y = 3x + 2 and y = 3x - 4 are shown on the grid. They never meet. What does this tell you about the solution of the simultaneous equations y = 3x + 2 and y = 3x - 4?
(Total for Question 9 is 1 mark)
10
The graphs of y = 1/2 x + 1 and y = -x + 7 are drawn on the grid. Write down the solution of the simultaneous equations y = 1/2 x + 1 and y = -x + 7.
(Total for Question 10 is 2 marks)
11
The graphs show the value of Sadia's and Tomasz's savings accounts over a number of weeks. Write down the number of weeks after which their savings are equal, and state the balance at that point.
(Total for Question 11 is 2 marks)
12
Which pair of simultaneous equations below has no solution, because the two lines are parallel?
A) y = 2x + 5 and y = 5x + 2
B) 3x + y = 8 and y = 3x - 8
C) 2y = 4x + 6 and y = 2x - 1
D) y = x + 4 and 2x - y = 4
(Total for Question 12 is 1 mark)
13
The graphs of y = -2x + 9 and y = 4x - 3 are drawn on the grid. Write down the solution of the simultaneous equations y = -2x + 9 and y = 4x - 3.
(Total for Question 13 is 2 marks)
14
The line y = x is already drawn on the grid.
(a)Complete the table of values for y = 9 - 2x. x: -1, 0, 1, 2, 3, 4 y: __, __, __, __, __, __(2)
(b)The line y = x is already drawn on the grid. Using your table, draw the graph of y = 9 - 2x on the same grid and hence write down the solution of the simultaneous equations y = 9 - 2x and y = x.(4)
(Total for Question 14 is 6 marks)
15
Solve the simultaneous equations x + y = 7 and y = 2x - 2 by drawing suitable straight-line graphs.
(a)Rearrange x + y = 7 to make y the subject.(1)
(b)By drawing suitable graphs on the grid, solve the simultaneous equations x + y = 7 and y = 2x - 2.(4)
(Total for Question 15 is 5 marks)
16
Without drawing any graphs, show that the simultaneous equations y = 4x - 3 and 8x - 2y = 7 do not have a solution.
(Total for Question 16 is 3 marks)
17
Two courier companies charge for delivering parcels. SwiftPost charges C = 2n + 30 and ParcelGo charges C = 5n + 9, where C is the cost in pounds and n is the number of parcels. The graphs of both charges, for n from 0 to 15, are shown on the grid.
(a)Write down the number of parcels for which the two companies charge the same amount, and state this cost.(2)
(b)A customer needs to send 15 parcels. By using the graphs or otherwise, determine which company is cheaper and find the difference in cost.(3)
(Total for Question 17 is 5 marks)
18
Line L1 passes through the points (0, 1) and (5, 11). Line L2 passes through the points (0, 10) and (6, 4).
(a)Line L1 passes through the points (0, 1) and (5, 11). Find the equation of L1 in the form y = mx + c.(2)
(b)Line L2 passes through the points (0, 10) and (6, 4). Find the equation of L2 in the form y = mx + c.(2)
(c)By plotting L1 and L2 on the grid, find the coordinates of the point where L1 and L2 intersect.(3)
(Total for Question 18 is 7 marks)
19
The lines y = kx + 4 and y = 3x - 2 intersect at the point where x = 3.
(a)The lines y = kx + 4 and y = 3x - 2 intersect at the point where x = 3. Find the y-coordinate of the point of intersection.(1)
(b)Hence find the value of k.(2)
(Total for Question 19 is 3 marks)
20
Line L has equation 2x + 3y = 12. Line M has equation y = kx - 4. Given that L and M intersect at a point where y = 2, find the value of k.
(Total for Question 20 is 3 marks)
21
Two lines have equations y = kx + 2 and y = 3x + 2, where k is a constant and k is not equal to 3.
(a)Show that the point (0, 2) lies on both lines for every value of k.(2)
(b)Explain why (0, 2) must be the only point of intersection of the two lines when k is not equal to 3.(2)
(Total for Question 21 is 4 marks)
Mark scheme · 5.12D Solving Simultaneous Equations Graphically: Fluency and Exam Drill
Question 1
B1 at least 3 of the 5 missing values correct
B1 all 5 missing values correct (cao)
Answer: x = -2, -1, 0, 1, 2, 3 gives y = -5, -3, -1, 1, 3, 5
Question 2
B1 x = 4
B1 y = 2 (oe as a coordinate pair)
Answer: x = 4, y = 2
Question 3
B1 x = -2
B1 y = -3 (oe as a coordinate pair)
Answer: x = -2, y = -3
Question 4
B1 x = 3
B1 y = 2 (oe as a coordinate pair)
Answer: x = 3, y = 2
Question 5
B1 x = 2
B1 y = 1 (oe as a coordinate pair)
Answer: x = 2, y = 1
Question 6
B1 B
Answer: B) x = -3, y = 4
Question 7
B1 y = 9 - 3x (oe)
Answer: y = 9 - 3x
Question 8
M1 substitutes x = 3 into both equations
A1 shows y = -1 from both equations, cso, so (3,-1) satisfies both equations
Answer: y = 2(3) - 7 = -1 and y = -3(3) + 8 = -1, both give y = -1, so (3,-1) satisfies both equations
Question 9
B1 there is no solution, because the lines are parallel (same gradient, different y-intercepts) so they never intersect
Answer: No solution - the lines are parallel and never meet
Question 10
B1 x = 4
B1 y = 3 (oe as a coordinate pair)
Answer: x = 4, y = 3
Question 11
B1 3 weeks
B1 balance = GBP 16
Answer: After 3 weeks, both accounts hold GBP16
Question 12
B1 C
Answer: C) 2y = 4x + 6 and y = 2x - 1
Question 13
B1 x = 2
B1 y = 5 (oe as a coordinate pair)
Answer: x = 2, y = 5
Question 14
(a) B1 at least 3 of the 6 values correct
(a) B1 all 6 values correct (cao)
(a) Answer: x = -1, 0, 1, 2, 3, 4 gives y = 11, 9, 7, 5, 3, 1
(b) M1 at least 4 points plotted correctly from the table, ft from (a)
(b) A1 correct ruled straight line y = 9 - 2x drawn from x = -1 to x = 4
(b) B1 identifies intersection at (3,3)
(b) B1 states solution x = 3, y = 3 (ft from their graph)
(b) Answer: x = 3, y = 3
Question 15
(a) B1 y = 7 - x (oe)
(a) Answer: y = 7 - x
(b) M1 at least 3 correct points for y = 7 - x, e.g. (0,7), (3,4), (6,1), ft from (a)
(b) M1 at least 3 correct points for y = 2x - 2, e.g. (0,-2), (2,2), (4,6)
(b) A1 both correct straight lines drawn
(b) B1 states solution x = 3, y = 4 (ft from their graphs, intersection identified)
(b) Answer: x = 3, y = 4
Question 16
M1 rearranges 8x - 2y = 7 to y = 4x - 3.5 (oe, e.g. y = 4x - 7/2)
A1 compares gradients and states both lines have gradient 4
A1 concludes the lines are parallel (different y-intercepts, -3 and -3.5) so the simultaneous equations have no solution
Answer: No solution - the two lines are parallel (both gradient 4, different y-intercepts)
Question 17
(a) B1 n = 7 parcels
(a) B1 cost = GBP 44
(a) Answer: n = 7 parcels, cost = GBP 44
(b) M1 substitutes n = 15 into both C = 2n + 30 and C = 5n + 9
(b) A1 SwiftPost = GBP 60 and ParcelGo = GBP 84
(b) A1 concludes SwiftPost is cheaper, by GBP 24 (ft from their two costs)
(b) Answer: SwiftPost is cheaper by GBP 24
Question 18
(a) M1 finds gradient = (11-1)/(5-0) = 2
(a) A1 y = 2x + 1 (oe), using y-intercept 1 from (0,1)
(a) Answer: y = 2x + 1
(b) M1 finds gradient = (4-10)/(6-0) = -1
(b) A1 y = -x + 10 (oe), using y-intercept 10 from (0,10)
(b) Answer: y = -x + 10
(c) M1 plots L1 correctly using the equation found in (a), ft
(c) M1 plots L2 correctly using the equation found in (b), ft
(c) A1 reads off intersection at (3, 7), ft from their graphs
(c) Answer: (3, 7)
Question 19
(a) B1 y = 7
(a) Answer: y = 7
(b) M1 substitutes the point (3, 7) into y = kx + 4
(b) A1 k = 1 (ft from (a))
(b) Answer: k = 1
Question 20
M1 substitutes y = 2 into 2x + 3y = 12 to find x = 3
M1 substitutes the point (3, 2) into y = kx - 4
A1 k = 2 (cao)
Answer: k = 2
Question 21
(a) M1 substitutes x = 0 into both y = kx + 2 and y = 3x + 2
(a) A1 shows both give y = 2 regardless of the value of k, cso, so (0,2) satisfies both equations
(a) Answer: y = k(0) + 2 = 2 and y = 3(0) + 2 = 2, both give y = 2 for every k, so (0,2) lies on both lines
(b) B1 since k is not equal to 3, the two lines have different gradients (k and 3), so they are not parallel and not identical
(b) B1 two distinct straight lines with different gradients intersect at exactly one point, and (a) shows this point is (0,2), so it must be the unique intersection
(b) Answer: Because k does not equal 3 the lines have different gradients, so they are not parallel; two non-parallel lines meet at exactly one point, and that point must be (0,2)