Equations and inequalities · Ages 13–14

Solving by graphs worksheet

Part of: Simultaneous linear equations

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Equations and inequalities · Ages 13–14

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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Solve ① x + 2y = −1 and ② 2x − 2y = 10 by the graphs: x = , y =
Fill in each blank
Answer:
2.Solve ① 2x + 2y = 14 and ② 3x − y = 5 by the graphs: x = , y =
Fill in each blank
Answer:
3.Lines ① and ② cross in the picture. With the crossing at (x, y), x + y =
Answer:
4.To draw 4x + 2y = 16, find two points first: (0, ) and (, 0).
Fill in each blank
Answer:
5.To draw x + 3y = −6, find two points first: (0, ) and (, 0).
Fill in each blank
Answer:
6.To draw 4x + 3y = 12, find two points first: (0, ) and (, 0).
Fill in each blank
Answer:

7.The picture shows lines ① and ②. Reading it, what is the solution (x, y) of the pair of equations?
- A. (1, 4)
- B. (4, −1)
- C. (4, 1)
- D. (−4, 1)
Choose the right answer
Answer:
8.Which pair of equations has graphs that are two parallel lines (no solution)?
- A. x + 2y = 6 and 2x + y = 6
- B. x + 2y = 6 and x − 2y = 6
- C. x + 2y = 6 and 2x + 4y = 14
- D. x + 2y = 6 and 2x + 4y = 12
Choose the right answer
Answer:
9.The picture shows that x = 4, y = −3 satisfies both ① and ②.
True or false?
Answer:
10.The picture shows that x = 5, y = −4 satisfies both ① and ②.
True or false?
Answer:
11.The lines 4x + 3y = 10 and 12x + 9y = 30 never meet.
True or false?
Answer:
12.The lines 4x + 4y = 6 and 12x + 12y = 18 never meet.
True or false?
Answer:

13.Solve the pair of equations shown by lines ① and ② using the graphs, giving x and y. xy =
Working:
Answer:
14.In the picture, line ① is −4x − 7y = 16 and ② is −4x − 6y = 12. They cross at A, and meet the x-axis at B and C. What is the area of triangle ABC, in square units?
Working:
Answer:
15.Before drawing 4x + 2y = 6, make a table: when x = 0, y = ; when x = 2, y = .
Fill in each blank
Working:
Answer:
16.Solving a pair of simultaneous equations by graphs, someone drew lines ① and ② in the picture. How many solutions does the pair have?
- A. Two
- B. Endlessly many
- C. Exactly one
- D. None
Choose the right answer
Working:
Answer:
17.In the picture, line ① is 2x + y = −5 and line ② is 3x − 2y = k. Reading the crossing: x = , y = ; so k = .
Fill in each blank
Working:
Answer:
18.In the picture, line ① is −2x − 6y = 8 and ② is −2x + y = −6. They cross at A, and meet the x-axis at B and C. What is the area of triangle ABC, in square units?
Working:
Answer:

Show the answers and working

Solving by graphs worksheet: Answers and working

1. 3, −2
The lines meet at (3, −2). Check: ①'s left side is −1 and ②'s is 10.
2. 3, 4
The lines meet at (3, 4). Check: ①'s left side is 14 and ②'s is 5.
3. −5
The crossing is (−3, −2), so x + y = −5.
4. 8, 4
With x = 0, 2y = 16, y = 8; with y = 0, 4x = 16, x = 4. Join the two points.
5. −2, −6
With x = 0, 3y = −6, y = −2; with y = 0, x = −6, x = −6. Join the two points.
6. 4, 3
With x = 0, 3y = 12, y = 4; with y = 0, 4x = 12, x = 3. Join the two points.

7. C. (4, 1)
Where the lines cross is on both, so its coordinates, (4, 1), are the solution.
8. C. x + 2y = 6 and 2x + 4y = 14
The coefficients of x and y are in proportion (2 times), but the constant isn't: the lines are parallel and never meet.
9. True
Only the crossing point, (4, −3), satisfies both equations.
10. False
Only the crossing point, (4, −4), satisfies both equations.
11. False
The second's x and y coefficients are 3 times the first's. If its constant is 3 times too, they are one line (meeting everywhere); if not, they are parallel (never meeting).
12. False
The second's x and y coefficients are 3 times the first's. If its constant is 3 times too, they are one line (meeting everywhere); if not, they are parallel (never meeting).

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. −2
The crossing is (2, −1), so xy = (2) × (−1) = −2.
14. 2 square units
A = (3, −4); with y = 0, B = (−4, 0) and C = (−3, 0). Base BC = −3 − (−4), height the distance from A to the x-axis, |−4|: area = 2.
15. 3, −1
x = 0: 2y = 6, y = 3; x = 2: 2y = 6 − 4 × 2, y = −1. Join the two points.
16. D. None
The number of crossings is the number of solutions: crossing, one; parallel, none; on top of each other, endlessly many.
17. −1, −3, 3
The crossing (−1, −3) is on ②: k = 3 × (−1) − 2 × (−3) = 3.
18. 7 square units
A = (2, −2); with y = 0, B = (−4, 0) and C = (3, 0). Base BC = 3 − (−4), height the distance from A to the x-axis, |−2|: area = 7.

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