CONCURRENCY OF THREE LINES WORKSHEET

In each, prove that the three lines are concurrent and also find the point of concurrency.

Problem 1 :

2x + y = 1

2x + 3y = 3

3x + 2y = 2

Problem 2 :

x + y - 5 = 0

3x - y + 1 = 0

5x - y - 1 = 0

Problem 3 :

3x + y = -2

2x - y = -3

x + 4y = 3 

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Answers

1. Answer :

2x + y = 1 ----(1)

2x + 3y = 3 ----(2)

3x + 2y = 2 ----(3)

Solve (1) and (2) to find the point of intersection.

Solve (1) for y in terms of x.

2x + y = 1

y = 1 - 2x ----(4)

Substitute y = 1 - 2x into (2).

2x + 3(1 - 2x) = 3

2x + 3 - 6x = 3

-4x + 3 = 3

-4x = 0

x = 0

Substitute x = 0 into (4).

y = 1 - 2(0)

y = 1 - 0

y = 1

The point of intersection (1) and (2) is (0, 1).

Substitute (0, 1) into (3).

3(0) + 2(1) = 2 (true ?)

0 + 2 = 2 (true ?)

2 = 2 (true)

The point (0, 1) satisfies (3).

Therefore, the given three lines are concurrent and the point of concurrency is (0, 1).

2. Answer :

x + y - 5 = 0 ----(1)

3x - y + 1 = 0 ----(2)

5x - y - 1 = 0 ----(3)

Solve (1) and (2) to find the point of intersection.

Solve (1) for y in terms of x.

x + y - 5 = 0

y = 5 - x ----(4)

Substitute y = 5 - x into (2).

3x - (5 - x) + 1 = 0

3x - 5 + x + 1 = 0

4x - 4 = 0

4x = 4

x = 1

Substitute x = 1 into (4).

y = 5 - 1

y = 4

The point of intersection (1) and (2) is (1, 4).

Substitute (1, 4) into (3).

5(1) - 4 - 1 = 0 (true ?)

5 - 4 - 1 = 0 (true ?)

0 = 0 (true)

The point (1, 4) satisfies (3).

Therefore, the given three lines are concurrent and the point of concurrency is (1, 4).

3. Answer :

3x + y = -2 ----(1)

2x - y = -3 ----(2)

x + 4y = 3 ----(3)

Solve (1) and (2) to find the point of intersection.

Solve (1) for y in terms of x.

3x + y = -2

y = -2 - 3x ----(4)

Substitute y = -2 - 3x into (2).

2x - (-2 - 3x) = -3

2x + 2 + 3x = -3

5x + 2 = -3

5x = -5

x = -1

Substitute x = -1 into (4).

y = -2 - 3(-1)

y = -2 + 3

y = 1

The point of intersection (1) and (2) is (-1, 1).

Substitute (-1, 1) into (3).

x + 4y = 3 (true ?)

-1 + 4(1) = 3 (true ?)

-1 + 4 = 3 (true ?)

3 = 3 (true)

The point (-1, 1) satisfies (3).

Therefore, the given three lines are concurrent and the point of concurrency is (-1, 1).

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