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:

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':
• $5x=2$, $x=\frac{2}{5}$
4. Finding 'y' using 'x':
• Substitute $x=\frac{2}{5}$ into Line 1 or Line 2 to find 'y'.
• Let's substitute into Line 1: $y=3\left(\frac{2}{5}\right)+2=\frac{6}{5}+2=\frac{16}{5}$
5. Conclusion:
• The point of intersection is $\left(\frac{2}{5},\frac{16}{5}\right)$.

Method 2: Elimination Method

1. Given Equations:
• $2x-3y=7$ (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:
• $2x-3y=7$
• $12x+3y=15$
4. Solve for 'x':
• Adding the equations gives $14x=22$, $x=\frac{22}{14}=\frac{11}{7}$
5. Finding 'y' using 'x':
• Substitute $x=\frac{11}{7}$ into Line 1: $2\left(\frac{11}{7}\right)-3y=7$
• Solving gives $y=-\frac{1}{7}$
6. Conclusion:
• The point of intersection is $\left(\frac{11}{7},-\frac{1}{7}\right)$.

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 ${m}_{1}=2$
• y-intercept ${c}_{1}=3$
2. Line 2 (y = -3x + 6):
• Slope ${m}_{2}=-3$
• y-intercept ${c}_{2}=6$

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':

• $2x+3x=6-3$
• $5x=3$
• $x=\frac{3}{5}$
3. Find 'y' using 'x':

• Substitute $x=\frac{3}{5}$ into Line 1 or Line 2 to find 'y'.
• Using Line 1: $y=2×\frac{3}{5}+3=\frac{6}{5}+3=\frac{21}{5}$

Conclusion:

The point of intersection between the lines $y=2x+3$ and $y=-3x+6$ is at $\left(\frac{3}{5},\frac{21}{5}\right)$.

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.