Systems of Equations
Two lines, one intersection. Or zero. Or infinite. Learn how to tell the difference instantly.
Foundation: What is a "System"?
A "system" is just math-speak for "Where do the lines cross?"
The Intersection Point (x, y)
The solution to a system is the only coordinate pair \((x,
y)\)
that works for BOTH equations at the same time.
If you plug the \(x\) and \(y\) back in, both equations must stay true.
The Three Outcomes
On the SAT, you rarely just "solve for x and y". Usually, they ask about the number of solutions.
One Solution
Visual: Intersecting Lines
Math: Different Slopes
y = -x + 3
No Solution
Visual: Parallel Lines
Math: Same Slope, Diff Y-Int
b₁ ≠ b₂
y = 2x - 1
Inf. Solutions
Visual: Same Line
Math: Same Slope, Same Y-Int
b₁ = b₂
2x + 2y = 10
Elimination vs. Substitution
Always use Elimination (adding/subtracting equations) unless one variable is already isolated (e.g., \(y = 3x + 1\)). Substitution is prone to sign errors.
Better yet: If it's a calculator section (which is the whole test now), just type them into Desmos.