Intersection of Two Lines Calculator

What is the Intersection of Two Lines?

The intersection of two lines is the point where they cross each other. Imagine two roads meeting at a corner - that's like the intersection of two lines in math!

How to Calculate the Intersection of Two Lines

We can find the intersection point by solving the equations of both lines together. There are two main ways to write line equations:

1. Slope-intercept form: \(y = ax + b\)
2. General form: \(Ax + By + C = 0\)

Formulas

For slope-intercept form (\(y = ax + b\)):

\[ \text{Line 1: } y = a_1x + b_1 \]

\[ \text{Line 2: } y = a_2x + b_2 \]

Where:

• \(a_1\) and \(a_2\) are the slopes of the lines
• \(b_1\) and \(b_2\) are the y-intercepts

For general form (\(Ax + By + C = 0\)):

\[ \text{Line 1: } A_1x + B_1y + C_1 = 0 \]

\[ \text{Line 2: } A_2x + B_2y + C_2 = 0 \]

Where \(A\), \(B\), and \(C\) are constants in each equation.

Calculation Steps

1. Write down the equations of both lines
2. Set the right sides of the equations equal to each other
3. Solve for x
4. Use the x-value in either equation to find y
5. The point (x, y) is the intersection!

Example and Visual Representation

Let's find where these two lines meet:

\[ \text{Line 1: } y = 2x + 1 \]

\[ \text{Line 2: } y = -x + 5 \]

Our calculator finds that they intersect at (4/3, 11/3) or approximately (1.33, 3.67)

Let's see this on a graph:

In this picture, the red line is y = 2x + 1, and the blue line is y = -x + 5. The green dot shows where they intersect at (1.33, 3.67).