A system of equations consists of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously — geometrically, this is where the graphs intersect.
The substitution method involves solving one equation for a variable and substituting that expression into the other equation. This works best when one equation already has a variable isolated. For example, if y = 2x + 1 and 3x + y = 11, substitute to get 3x + (2x + 1) = 11.
The elimination method adds or subtracts equations to eliminate a variable. Multiply one or both equations by constants so that adding them cancels one variable. For example, x + y = 5 and x - y = 1 add to give 2x = 6, so x = 3.
Systems can have one solution (the lines intersect), no solution (the lines are parallel), or infinitely many solutions (the lines are identical). The number of solutions depends on whether the equations are consistent and independent.