# Intersection of Two Lines: Formula, Methods, and Examples

TL;DR  
The intersection of two lines is the single point where they cross, found by solving both line equations together. This guide covers the substitution method, the determinant formula for general-form lines, how to spot parallel lines that never meet, and the mistakes students make most.

## What Is the Intersection of Two Lines?  
The intersection of two lines is the point (x,y) that lies on both lines at once. Because that point satisfies both equations, finding it means solving the two equations together as a [system of equations](/content/math/algebra/system-of-equations/index.html). Two straight lines in a plane can do one of three things: cross at exactly one point, run parallel and never meet, or sit on top of each other as the same line.

The key idea: the **point of intersection is the shared solution** of both line equations. It is where the two lines agree.

## How to Find the Point of Intersection  
**How do you find the point where two lines intersect?** The most reliable method works straight from slope-intercept form: make the two equations equal, solve for one coordinate, then back-substitute for the other.

When both lines are written as y=mx+b:

1. Set the two right-hand sides equal, since both equal y at the crossing point.  
2. Solve that single equation for x.  
3. Substitute that x back into either original line to get y.

The pair (x,y) is the point of intersection.

There is also a direct formula for lines given in general form, a1x+b1y+c1=0 and a2x+b2y+c2=0:

x=\frac{b1c2−b2c1}{a1b2−a2b1}, \qquad y=\frac{a2c1−a1c2}{a1b2−a2b1}

Notice the shared denominator a1b2−a2b1. **If that denominator is 0, the lines are parallel** and there is no single intersection point.

## Examples of Intersection of Two Lines  
These run from a clean substitution to the general-form formula and a parallel-line case. Each problem statement is bold; the steps are plain.

### Example 1  
**Find where y = x + 3 and y = 2x + 1 cross.**  
Set the right-hand sides equal:  
x+3=2x+1  
Solve for x:  
3−1=2x−x  
x=2  
Substitute x=2 into y=x+3:  
y=2+3=5  
Final answer: the lines cross at (2,5).

### Example 2  
**Find where y = 3x + 4 and y = 3x - 2 cross.**  
Your first instinct is to set them equal and solve as usual:  
3x+4=3x−2  
Subtract 3x from both sides:  
4=−2  
Final answer: no point of intersection; the lines are parallel.

### Example 3  
**Find where 2x + y = 5 and x - y = 1 cross.**  
Add the two equations to eliminate y:  
(2x+y)+(x−y)=5+1  
3x=6  
x=2  
Substitute x=2 into x−y=1:  
2−y=1  
y=1  
Final answer: the lines cross at (2,1).

### Example 4  
**Use the general-form formula to find where 3x + 2y - 4 = 0 and 2x - y - 5 = 0 cross.**  
Read off a1=3, b1=2, c1=−4 and a2=2, b2=−1, c2=−5. The denominator is:  
a1b2−a2b1=(3)(−1)−(2)(2)=−7  
Now x:  
x=\frac{b1c2−b2c1}{-7}  
And y:  
y=\frac{a2c1−a1c2}{-7}  
Final answer: the lines cross at (2,−1).

### Example 5  
**A line through (0, 1) with slope 1 meets a line through (0, 7) with slope -1. Where do they cross?**  
Write each in point-slope form, then simplify: y=x+1 and y=−x+7. Set them equal:  
x+1=−x+7  
Final answer: the lines cross at (3,4).

### Example 6  
**A delivery scooter's position is y = 30x (km after x hours) and a competitor's is y = 20x + 10. When and where are they level?**  
Set the positions equal:  
30x=20x+10  
Final answer: they are level at (1,30).

## Why Intersection Matters: "Where Two Conditions Meet at Once"  
Solving for an intersection is solving two truths at the same time. Each line is a rule, and the crossing point is the one place where both rules hold. That is why the idea shows up far beyond geometry class.  
- **Break-even points.** In business, one line is cost and another is revenue. Their intersection is the break-even point.  
- **Navigation and positioning.** Two bearings from two known landmarks give a line; where they cross is your position.  
- **Meeting and collision.** Two objects moving along straight paths meet at the intersection.

## Common Mistakes With Intersection of Two Lines  
### Mistake 1:  Forgetting to find the second coordinate  
**Don't do this:** Reporting "x=2" as the point of intersection.  
**The correct way:** Report the complete point, (x,y).

### Mistake 2: Not checking for parallel lines first  
**The correct way:** Compare slopes before solving.

### Mistake 3: Substituting into a wrong or rearranged equation  
**The correct way:** Substitute x back into one of the **original** equations.

## Conclusion  
- The **intersection of two lines** is the single point that satisfies both line equations.  
- An intersection is a point (x,y), so solve for both coordinates.  
- Equal slopes mean parallel lines with no intersection, or the same line if intercepts also match.