Systems of equations
Also known as: simultaneous equations, system of linear equations
A system of equations is a set of two or more equations sharing the same variables, solved together to find values that satisfy every equation at once. The solution to a two-variable linear system is the point where the lines intersect.
A single equation with two unknowns has infinitely many solutions. Pairing it with a second equation pins the answer down: the solution to the system is the ordered pair (x, y) that makes both equations true simultaneously. Graphically, each linear equation is a line, and the solution is their point of intersection. As a rule, you need as many independent equations as you have variables.
Two standard algebraic methods solve linear systems. Substitution solves one equation for a variable and plugs that expression into the other — efficient when a variable already has a coefficient of 1. Elimination adds or subtracts multiples of the equations to cancel a variable outright. To solve 2x + y = 11 and x − y = 1 by elimination, add them to get 3x = 12, so x = 4, then substitute back to find y = 3.
Not every system has one solution. If the two lines are parallel — the same slope but different intercepts — the system is inconsistent and has no solution; elimination produces a false statement like 0 = 5. If the two equations describe the same line, the system is dependent with infinitely many solutions, and elimination produces 0 = 0. Recognizing those outcomes from the coefficients alone is a common shortcut.
Systems of equations show up on nearly every quantitative admissions and placement test. The GRE quantitative section, the ASVAB Algebra II material, and the CLT algebra section all include them, often as word problems where the hardest step is translating two real-world conditions into two equations before any algebra begins.
Key takeaways
- A system of equations is solved by finding values that satisfy all equations at once.
- For two linear equations, the solution is the intersection point of the two lines.
- Substitution and elimination are the standard algebraic solution methods.
- Parallel lines give no solution (inconsistent); identical lines give infinitely many (dependent).
- You generally need as many independent equations as unknowns to reach a single solution.