A system of equations is a collection of two or more equations that share the same variables, and its solution is the set of values that satisfies every equation in the system at the same time. There are three principal methods for solving a system of two linear equations. The graphing method finds the point where the two lines intersect on a coordinate plane. The substitution method solves one equation for one variable in terms of the others and then plugs that expression into the remaining equation. The elimination method adds or subtracts the equations from each other in order to cancel one of the variables.
The substitution method works well when one equation is already solved, or easily solved, for a variable. Given the system \(y = 2x + 1\) and \(3x + y = 11\), you substitute \(2x + 1\) for \(y\) in the second equation to obtain \(3x + (2x + 1) = 11\), which simplifies to \(5x + 1 = 11\), so \(x = 2\). Substituting back gives \(y = 2(2) + 1 = 5\), and the solution is the ordered pair \((2, 5)\). The elimination method, by contrast, is most efficient when the coefficients of one variable are opposites or can be made so. In the system \(2x + 3y = 12\) and \(4x - 3y = 6\), adding the equations eliminates \(y\) and yields \(6x = 18\), so \(x = 3\). Substituting back gives \(2(3) + 3y = 12\), or \(3y = 6\), and therefore \(y = 2\); the solution is \((3, 2)\).
Not every system has a unique solution. A system is called inconsistent when it has no solution, which graphically corresponds to two parallel lines that never cross; algebraically, this situation arises when the equations reduce to a contradiction. A system is called dependent when it has infinitely many solutions, which happens when the two equations actually describe the same line so that every point on the line is a solution. You can predict the number of solutions for a \(2 \times 2\) linear system by comparing the slopes and y-intercepts of its lines: different slopes give one solution, equal slopes with different intercepts give no solution, and equal slopes with equal intercepts give infinitely many solutions.
For a quick algebraic approach to a \(2 \times 2\) system, Cramer's Rule uses determinants. For the system \(ax + by = e\) and \(cx + dy = f\), the solutions are \(x = \frac{ed - bf}{ad - bc}\) and \(y = \frac{af - ce}{ad - bc}\), provided the determinant \(ad - bc\) is not zero. A zero determinant signals either no solution or infinitely many solutions, mirroring the parallel-line and same-line cases.