Use the graphs to solve the simultaneous equations 2 – 2y = x and 2y = 3x – 22.
Use these graphs to solve the simultaneous equations
2 - 2y = x
2y = 3x - 22
2 - 2y = x
2y = 3x - 22
Sim. equ. graphically — Notes & Worked Examples Show / Hide
Explanation:
To solve simultaneous equations graphically, plot both equations on the same set of axes and find the coordinates (x, y) of the point(s) where the graphs intersect. At an intersection point, both equations share the exact same x-value and y-value.
Two non-parallel straight lines cross at one point (one pair of solutions). A straight line and a quadratic curve can cross twice (two pairs of solutions), touch once as a tangent, or never meet (no real solutions).
Example 1: The lines y = 2x − 1 and x + y = 8 are drawn on a grid and cross at (3, 5). Write down the solution to the simultaneous equations.
Solution:
Read intersection coordinates:
x = 3, y = 5
Check: 2(3)−1 = 5; 3+5 = 8 ✓
Answer: x = 3, y = 5
Example 2: The curve y = x2 − 3 and the line y = x + 3 intersect at (−2, 1) and (3, 6). Solve the simultaneous equations graphically.
Solution:
Read both intersections:
• (−2, 1) → x = −2, y = 1
• (3, 6) → x = 3, y = 6
Ans: (−2, 1) or (3, 6)
(a) Use the graphs to solve 5x – 9y = –46 and y = –2x (1)
(a) Use these graphs to solve the simultaneous equations
5x - 9y = -46
y = -2x [1 mark]
5x - 9y = -46
y = -2x [1 mark]
Worked Solution & Answer
(a)
Lines intersect at (–2, 4) B1
Answer (a): x = –2, y = 4
The graphs with equations 3y + 2x = 12 and 2y - 3x = -11312 have been drawn on the grid below.
Using the graphs, find estimates of the solutions of the simultaneous equations
3y + 2x = 12
2y - 3x = -11312
3y + 2x = 12
2y - 3x = -11312
Worked Solution & Answer
Intersection of the two lines on the grid is at x ≈ 2.25 (or 2.2 to 2.3), y ≈ –1.33 (or –1.3 to –1.4) M1 A1
Answer: x = 2.25, y = –1.33
Here is the graph of the equation 2y + x = 1 drawn on a grid.
By drawing another straight line on the grid, solve the simultaneous equations
y - x - 2 = 0
2y + x = 1
y - x - 2 = 0
2y + x = 1
Worked Solution & Answer
Draw line y = x + 2 through (−2, 0), (0, 2), (2, 4) M2
Intersection with 2y + x = 1 is at x = −1, y = 1. A1
Intersection with 2y + x = 1 is at x = −1, y = 1. A1
Answer: x = −1, y = 1
The graph shows the solution to this pair of simultaneous equations.
2x + y = 14
x + 2y = 13
Use the solution to work out the value of 3x + y.
You must show how you work out your answer.
x + 2y = 13
Use the solution to work out the value of 3x + y.
You must show how you work out your answer.
3x + y =
Worked Solution & Answer
The lines intersect at (5, 4), so x = 5 and y = 4. M2
Substitute into 3x + y: 3(5) + 4 = 15 + 4 = 19. A1
Substitute into 3x + y: 3(5) + 4 = 15 + 4 = 19. A1
Answer: 3x + y = 19
The graph of the line y = 2 - 12x is shown on this grid.
By drawing a suitable line on the grid, solve these simultaneous equations.
y = 2 - 12x and y = x - 1
y = 2 - 12x and y = x - 1
Worked Solution & Answer
Draw the straight line y = x − 1 passing through (0, −1), (1, 0), (2, 1), (4, 3) M3
The lines intersect at (2, 1) ⇒ x = 2, y = 1. A1
The lines intersect at (2, 1) ⇒ x = 2, y = 1. A1
Answer: x = 2, y = 1
A pattern is made from four identical rectangles.
The sides of the rectangles are parallel to the axes.
The sides of the rectangles are parallel to the axes.
Point A has coordinates (3, 4)
Point B has coordinates (11, 20)
Point C is marked on the diagram.
Work out the coordinates of C.
You must show all your working.
Point B has coordinates (11, 20)
Point C is marked on the diagram.
Work out the coordinates of C.
You must show all your working.
Worked Solution & Answer
Let length = L and width = W M2
Horizontal change from A(3,4) to B(11,20): 2L – W = 11 – 3 = 8. Vertical change from A(3,4) to B(11,20): 2L + W = 20 – 4 = 16 M2
Adding gives 4L = 24 ⇒ L = 6, W = 4. Therefore C = (3 + L, 4 + W) = (3 + 6, 4 + 4) = (9, 8). A1
Horizontal change from A(3,4) to B(11,20): 2L – W = 11 – 3 = 8. Vertical change from A(3,4) to B(11,20): 2L + W = 20 – 4 = 16 M2
Adding gives 4L = 24 ⇒ L = 6, W = 4. Therefore C = (3 + L, 4 + W) = (3 + 6, 4 + 4) = (9, 8). A1
Answer: (9, 8)
The diagrams show the price paid by two groups of people visiting a funfair.
Assume each adult pays the same price and each child pays the same price.
Find the price for an adult and the price for a child.
Find the price for an adult and the price for a child.
Worked Solution & Answer
5a + 4c = 78 (×3 ⇒ 15a + 12c = 234) M2
3a + 6c = 63 (×2 ⇒ 6a + 12c = 126) M1
Subtract: 9a = 108 ⇒ a = £12 M1
Substitute a = 12: 5(12) + 4c = 78 ⇒ 4c = 18 ⇒ c = £4.50. A1
3a + 6c = 63 (×2 ⇒ 6a + 12c = 126) M1
Subtract: 9a = 108 ⇒ a = £12 M1
Substitute a = 12: 5(12) + 4c = 78 ⇒ 4c = 18 ⇒ c = £4.50. A1
Answer: Adult price = £12, Child price = £4.50
(c) On the same grid, draw the graph of y = 2x - 6 for -1 ≤ x ≤ 5.
(d) Use your graphs to solve the equation x2 - 4x + 1 = 2x - 6.
Give your answers to 1 decimal place.
(d) Use your graphs to solve the equation x2 - 4x + 1 = 2x - 6.
Give your answers to 1 decimal place.
(c) On the same grid, draw the graph of y = 2x - 6 for -1 ≤ x ≤ 5. [3 marks]
(d) Use your graphs to solve the equation x2 - 4x + 1 = 2x - 6.
Give your answers to 1 decimal place. [2 marks]
Give your answers to 1 decimal place. [2 marks]
x =
Worked Solution & Answer
(c)
Plot (−1, −8), (0, −6), (3, 0), (5, 4) on the grid and join with a ruled straight line. M2 A1
Answer (c): Straight line drawn from (−1, −8) through (3, 0) to (5, 4)
(d)
Read the x-coordinates of the points of intersection of y = x² − 4x + 1 and y = 2x − 6: M1
x = 1.6 and x = 4.4 (or 1.5–1.7 and 4.3–4.5). A1
x = 1.6 and x = 4.4 (or 1.5–1.7 and 4.3–4.5). A1
Answer (d): x = 1.6 or x = 4.4
The graph of the line x + y = 3 is shown on this grid.
(a) By drawing a suitable line on the grid, solve these simultaneous equations.
x + y = 3 and y = 2x - 3 [4 marks]
x + y = 3 and y = 2x - 3 [4 marks]
(b) Umi says,
The simultaneous equations y = 2x + 3 and 2y - 4x = 7 will have no solutions.
Is Umi correct?
Explain how you decide. [2 marks]
The simultaneous equations y = 2x + 3 and 2y - 4x = 7 will have no solutions.
Is Umi correct?
Explain how you decide. [2 marks]
Worked Solution & Answer
(a)
Draw the ruled line y = 2x − 3 passing through (0, −3), (2, 1), (4, 5) M3
Point of intersection with x + y = 3 is (2, 1) ⇒ x = 2, y = 1. A1
Point of intersection with x + y = 3 is (2, 1) ⇒ x = 2, y = 1. A1
Answer (a): x = 2, y = 1
(b)
2y − 4x = 7 ⇒ y = 2x + 3.5. M1
Both lines have the same gradient (2) and different y-intercepts, so they are parallel and never intersect. C1
Both lines have the same gradient (2) and different y-intercepts, so they are parallel and never intersect. C1
Answer (b): Umi is correct because 2y − 4x = 7 rearranges to y = 2x + 3.5, so both lines have gradient 2 and are parallel.