Systems of two linear equations in two variables
A system of two linear equations has one solution when both lines meet at one point, and no or infinitely many solutions otherwise.
Core Idea
A system of two linear equations asks you to find the one pair
Understanding
A system is two lines that must agree at one point. Sometimes that point exists once, sometimes not at all, and sometimes every point works.
- Crossing lines: one solution.
- Parallel lines: no solution.
- Same line: infinitely many solutions.
On the SAT, you solve by substitution or elimination and then interpret the result in a graph or context. Pick the method that cancels fastest, then check the answer back in the original equations.
Concept Guides
5Solve systems using substitution and elimination.
Use substitution or elimination, whichever cancels faster.
Interpret the solution as the intersection point of two lines.
The solution to a system is the point where the two lines intersect.
Determine whether a system has one solution, no solution, or infinitely many solutions.
Use the equations or the line slopes to tell whether a system has one, none, or infinitely many solu
Connect systems to graphs and match a system to its graphical representation.
Match slopes, intercepts, and the intersection point to the graph of the system.
Build a system from a word problem and interpret the solution.
Write one equation per condition, solve the system, and interpret the answer in context.