Point of Intersection of Two Lines

Introduction:

  • The point of intersection between two lines represents the coordinates where the lines cross each other on a Cartesian plane.

Equations of Lines:

  • Two lines can be represented by equations such as:
    • Line 1: y=m1x+c1 or Ax+By+C1=0
    • Line 2: y=m2x+c2 or Dx+Ey+C2=0

Methods to Find Intersection:

  1. Substitution Method:

    • Solve the system of equations formed by the two line equations to find the x and y coordinates.
    • Substitute the solution into either of the line equations to confirm the point of intersection.
  2. Elimination Method:

    • Manipulate the equations to eliminate one variable (either x or y).
    • Solve for the remaining variable and then substitute the value into one of the line equations to find the other variable.
    • Confirm the point of intersection by substituting values into the second line equation.
  3. Using Slopes and Intercepts:

    • Equate the two line equations and solve for x to find the x-coordinate of the intersection point.
    • Substitute this x-value into one of the line equations to obtain the corresponding y-coordinate.

Method 1: Substitution Method

  1. Given Equations:
    • y=3x+2 (Line 1)
    • y=2x+4 (Line 2)
  2. Setting Equations Equal to Each Other:
    • 3x+2=2x+4
  3. Solving for 'x':
    • 5x=2, x=25
  4. Finding 'y' using 'x':
    • Substitute x=25 into Line 1 or Line 2 to find 'y'.
    • Let's substitute into Line 1: y=3(25)+2=65+2=165
  5. Conclusion:
    • The point of intersection is (25,165).

Method 2: Elimination Method

  1. Given Equations:
    • 2x3y=7 (Line 1)
    • 4x+y=5 (Line 2)
  2. Eliminate a Variable:
    • Multiply Line 2 by 3 to make the 'y' coefficients cancel out: 12x+3y=15
  3. Combine Equations:
    • 2x3y=7
    • 12x+3y=15
  4. Solve for 'x':
    • Adding the equations gives 14x=22, x=2214=117
  5. Finding 'y' using 'x':
    • Substitute x=117 into Line 1: 2(117)3y=7
    • Solving gives y=17
  6. Conclusion:
    • The point of intersection is (117,17).

Example:

Given two lines:

  • Line 1: y=2x+3
  • Line 2: y=3x+6

Using Slopes and Intercepts Method:

Step 1: Identify Slopes and Intercepts

  1. Line 1 (y = 2x + 3):
    • Slope m1=2
    • y-intercept c1=3
  2. Line 2 (y = -3x + 6):
    • Slope m2=3
    • y-intercept c2=6

Step 2: Find Intersection

  1. Equating Lines:

    • Set the equations of the lines equal to each other:
    • 2x+3=3x+6
  2. Solve for 'x':

    • 2x+3x=63
    • 5x=3
    • x=35
  3. Find 'y' using 'x':

    • Substitute x=35 into Line 1 or Line 2 to find 'y'.
    • Using Line 1: y=2×35+3=65+3=215

Conclusion:

The point of intersection between the lines y=2x+3 and y=3x+6 is at (35,215).

Importance:

Importance:

  • Intersection points help understand relationships between lines.
  • They are crucial in various fields like engineering, physics, and computer science for solving problems involving intersecting paths or systems.

Summary:

  • Methods like substitution and elimination facilitate finding the point of intersection between lines.
  • Verifying the obtained coordinates by substituting into original equations confirms the accuracy of the point of intersection.