# Condition of Concurrency of Straight Lines

#### Introduction:

- Concurrency of lines refers to the condition where three or more lines intersect at a single point.

#### Condition for Concurrency:

- In a Cartesian plane, three lines are concurrent if and only if the point of intersection of any two lines also lies on the third line.

#### Theorem of Concurrency:

**Three lines represented by equations:**- Line 1: $Ax+By+{C}_{1}=0$
- Line 2: $Px+Qy+{C}_{2}=0$
- Line 3: $Mx+Ny+{C}_{3}=0$

**Condition for Concurrency:**- If the point of intersection of Line 1 and Line 2 also lies on Line 3 (or vice versa), then the three lines are concurrent.

#### Method to Check Concurrency:

**Step 1: Find Intersections:**- Find the point of intersection between Line 1 and Line 2 using simultaneous equations.

**Step 2: Verify the Third Line:**- Substitute the obtained coordinates into the equation of the third line.
- If the coordinates satisfy the equation, the lines are concurrent.

### Example:

Given three lines:

- $2x-3y+4=0$
- $4x+y-6=0$
- $6x-2y+8=0$

### Checking for Concurrency:

#### Step 1: Finding Intersection of Line 1 and Line 2:

- Solve equations 1 and 2 to find the point of intersection.
- Equation 1: $2x-3y+4=0$
- Equation 2: $4x+y-6=0$

Solving these equations simultaneously gives:

- $2x-3y+4=0$ ...(1)
- $4x+y-6=0$ ...(2)

Multiply equation (2) by 3 to eliminate 'y':

$12x+3y-18=0$ ...(3)

Add equations (1) and (3):

$14x-14=0$

$14x=14$

$x=1$

Substitute $x=1$ into equation (1):

$2(1)-3y+4=0$

$2-3y+4=0$

$-3y+6=0$

$-3y=-6$

$y=2$

Hence, the point of intersection for Line 1 and Line 2 is $(1,2)$.

#### Step 2: Verifying Line 3:

- Equation 3: $6x-2y+8=0$
- Substitute the obtained intersection point $(1,2)$ into the equation of Line 3:

$6(1)-2(2)+8=6-4+8=10\mathrm{\ne}0$

### Conclusion:

As the coordinates $(1,2)$ do not satisfy the equation of Line 3, the lines $2x-3y+4=0$, $4x+y-6=0$, and $6x-2y+8=0$are not concurrent.

#### Importance:

- Concurrency of lines signifies a significant intersection point where multiple relationships and geometrical properties can be derived.
- It's crucial in various fields such as architecture, engineering, and mathematics to understand the intersection and relationships between multiple lines.