How To Solve A System Of Linear Equations

A system of linear equations is a set of equations that must all be true at the same time. Two equations in two unknowns describe two straight lines, and the solution is the point where those lines cross. Written in the standard form ax + by = c, the pair has exactly one solution unless the lines are parallel or identical.

Cramer's rule solves the system by determinants rather than by eliminating one variable at a time. The determinant of the coefficients measures how far apart the two lines are in direction, and it appears in the denominator of both answers. When it is zero the lines are parallel or the same line, and there is no single crossing point.

The coefficient determinant is built from the four numbers that multiply the unknowns: a1 b2 minus a2 b1. For the pair 2x + 3y = 13 and 4x - y = 5 that is 2 times -1 minus 4 times 3, which is -14. The sign matters, and a determinant is allowed to be negative.

The numerator for x is formed by replacing the x coefficients with the constants, keeping the y coefficients where they are. That gives c1 b2 minus c2 b1, which for the example is 13 times -1 minus 5 times 3, or -28. Dividing by the determinant gives 2.

The numerator for y replaces the y coefficients with the constants instead, giving a1 c2 minus a2 c1. For the example that is 2 times 5 minus 4 times 13, or -42, and dividing by -14 gives 3. The symmetry between the two numerators is what makes the rule easy to remember once the pattern is clear.

A zero determinant means the two lines are parallel and therefore never meet, or that they are the same line and meet everywhere. In the first case the system is inconsistent and has no solution; in the second it has infinitely many. Either way there is no single pair of numbers to report, which is why the determinant has to be checked before dividing.

The quick way to spot the parallel case is to compare ratios. If the ratio of the x coefficients equals the ratio of the y coefficients, the lines have the same slope. If the ratio of the constants differs as well, they are parallel and distinct; if it matches too, the equations are the same line written twice.

Substituting the answer back into the original equations is the check that costs nothing and catches almost every arithmetic slip. A solution that satisfies one equation but not the other means the determinant or one of the numerators was built with the wrong coefficient. Both equations must return their stated constants.

The method extends to three equations in three unknowns, where the determinant is a three by three array and each numerator replaces one column with the constants. Beyond three the arithmetic grows quickly, and elimination methods or matrix factorisation take over. For a two by two system, Cramer's rule is about as direct as it gets.

Graphing the two lines makes the meaning of the determinant concrete. Two lines with different slopes cross once, and the determinant is the quantity that encodes the difference in slope. Two lines with the same slope never cross unless they are identical, and that is exactly when the determinant vanishes.