Intersection of Two Lines
Master finding where two lines meet — a core SAT Math skill
Algebra
Systems of Equations
SAT Core Topic
1. What Is an Intersection?
Definition
The intersection of two lines is the point where they cross. It is the single location $(x, y)$ that lies on both lines simultaneously.
Three Perspectives:
- Geometric: The point where two lines cross on a graph
- Algebraic: The $(x, y)$ values that satisfy both equations at the same time
- Real-World: The moment when two quantities are equal (break-even, meeting point, etc.)
Why It Matters on the SAT
- Solving systems of linear equations
- Break-even and comparison word problems
- Interpreting graphs of two functions
- Determining when two situations are equal
Interactive visualization — available in the full lesson
The green point $(3, 7)$ satisfies both equations
2. Three Possible Cases
Exactly One Intersection
Most Common on SAT
- Slopes are different: $m_1 \neq m_2$
- Lines cross at exactly one point
- System has a unique solution
- Called an independent, consistent system
Algebraic signal: You get a specific value for $x$ and $y$
No Intersection
Parallel Lines
- Same slope, different intercepts: $m_1 = m_2$, $b_1 \neq b_2$
- Lines never cross
- System has no solution
- Called an inconsistent system
Algebraic signal: Variables cancel → false statement like $0 = 5$
Infinite Intersections
Same Line
- Same slope AND same intercept: $m_1 = m_2$, $b_1 = b_2$
- Equations describe the identical line
- System has infinitely many solutions
- Called a dependent system
Algebraic signal: Variables cancel → true statement like $0 = 0$
Interactive visualization — available in the full lesson
Interactive visualization — available in the full lesson
Interactive visualization — available in the full lesson