Before you read

Answer first — see what you already know.

Two lines are graphed: $y = x + 1$ and $y = - x + 5$ . What is their intersection point?

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Answer:

Before you read

Answer first — see what you already know.

The graphs of $y = 3 x - 2$ and $y = 3 x + 4$ are shown. Based on the graphs, how many solutions does this system have?

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Answer:

Theory

Systems as Intersecting Lines

The solution to a system of two linear equations is the point where the two lines cross on a graph.

The solution is where the lines cross: (2, 2)

To find it graphically:
1. Graph both lines
2. Find the intersection point
3. Read the coordinates $(x, y)$

With Desmos on the SAT, this is incredibly fast:
- Type both equations
- Click on the intersection point
- Read the answer

This method is sometimes faster than algebra, especially for messy systems!

Theory

What the Graph Tells You

Graph appearance	Number of solutions	Algebraic result
Lines cross at one point	One solution	You find $(x, y)$
Lines are parallel (same slope, different intercept)	No solution	You get $0 = 5$ (false)
Lines overlap completely (same line)	Infinitely many	You get $0 = 0$ (true)

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