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: $y={m}_{1}x+{c}_{1}$ or $Ax+By+{C}_{1}=0$
 Line 2: $y={m}_{2}x+{c}_{2}$ or $Dx+Ey+{C}_{2}=0$
Methods to Find Intersection:

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.

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.

Using Slopes and Intercepts:
 Equate the two line equations and solve for x to find the xcoordinate of the intersection point.
 Substitute this xvalue into one of the line equations to obtain the corresponding ycoordinate.
Method 1: Substitution Method
 Given Equations:
 $y=3x+2$ (Line 1)
 $y=2x+4$ (Line 2)
 Setting Equations Equal to Each Other:
 $3x+2=2x+4$
 Solving for 'x':
 $5x=2$, $x=\frac{2}{5}$
 Finding 'y' using 'x':
 Substitute $x=\frac{2}{5}$ into Line 1 or Line 2 to find 'y'.
 Let's substitute into Line 1: $y=3(\frac{2}{5})+2=\frac{6}{5}+2=\frac{16}{5}$
 Conclusion:
 The point of intersection is $(\frac{2}{5},\frac{16}{5})$.
Method 2: Elimination Method
 Given Equations:
 $2x3y=7$ (Line 1)
 $4x+y=5$ (Line 2)
 Eliminate a Variable:
 Multiply Line 2 by 3 to make the 'y' coefficients cancel out: $12x+3y=15$
 Combine Equations:
 $2x3y=7$
 $12x+3y=15$
 Solve for 'x':
 Adding the equations gives $14x=22$, $x=\frac{22}{14}=\frac{11}{7}$
 Finding 'y' using 'x':
 Substitute $x=\frac{11}{7}$ into Line 1: $2(\frac{11}{7})3y=7$
 Solving gives $y=\frac{1}{7}$
 Conclusion:
 The point of intersection is $(\frac{11}{7},\frac{1}{7})$.
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
 Line 1 (y = 2x + 3):
 Slope ${m}_{1}=2$
 yintercept ${c}_{1}=3$
 Line 2 (y = 3x + 6):
 Slope ${m}_{2}=3$
 yintercept ${c}_{2}=6$
Step 2: Find Intersection

Equating Lines:
 Set the equations of the lines equal to each other:
 $2x+3=3x+6$

Solve for 'x':
 $2x+3x=63$
 $5x=3$
 $x=\frac{3}{5}$

Find 'y' using 'x':
 Substitute $x=\frac{3}{5}$ into Line 1 or Line 2 to find 'y'.
 Using Line 1: $y=2\times \frac{3}{5}+3=\frac{6}{5}+3=\frac{21}{5}$
Conclusion:
The point of intersection between the lines $y=2x+3$ and $y=3x+6$ is at $(\frac{3}{5},\frac{21}{5})$.
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.