Math
Find the x and y that satisfy two linear equations at once — by the determinant method, with both equations checked.
Equation 1: a₁x + b₁y = c₁ · Equation 2: a₂x + b₂y = c₂
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For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, let D = a₁b₂ − a₂b₁. If D ≠ 0, there is exactly one solution: x = (c₁b₂ − c₂b₁) ÷ D and y = (a₁c₂ − a₂c₁) ÷ D. This is Cramer's rule for two equations. If D = 0, the lines are parallel or identical: parallel lines never meet (no solution), identical lines meet everywhere (infinitely many solutions).
2x + y = 7 and x − y = 2: D = 2 × (−1) − 1 × 1 = −3. Then x = (7 × (−1) − 2 × 1) ÷ (−3) = (−9) ÷ (−3) = 3, and y = (2 × 2 − 1 × 7) ÷ (−3) = (−3) ÷ (−3) = 1. Check: 2 × 3 + 1 = 7, and 3 − 1 = 2. Both equations hold, so (3, 1) is the single crossing point of the two lines.
This page solves two linear equations in two unknowns. It does not handle three or more equations, nonlinear terms (x², xy), or inequalities. When D = 0 it distinguishes the two cases: if the equations are proportional (same line), it reports infinitely many solutions; otherwise it reports no solution. Results display to two decimals; repeating decimals such as 1/3 show as 0.33, so keep the fraction form for exact work.
Each equation is a straight line, and the solution is their crossing point. One crossing means one solution; parallel lines never cross (no solution); the same line twice means every point on it works (infinitely many solutions).
Then the two lines have the same slope. The page checks proportionality: if one equation is a multiple of the other, the lines coincide and every point on the line is a solution; otherwise they are distinct parallels with no solution at all.
After Gabriel Cramer, who published the determinant method in 1750. It is the same elimination you would do by hand — multiply, subtract, solve — compressed into two formulas.
Yes, and you should get the same answer. From x − y = 2, x = y + 2; substituting into 2x + y = 7 gives 2(y + 2) + y = 7, so 3y = 3, y = 1, x = 3. Substitution is easier by hand for simple systems; determinants scale better.
Yes for this method. If either equation contains x², y², xy, or a square root of x or y, the determinant formulas do not apply, and the system can have zero, one, two, or more solutions.
Substitute x and y back into both original equations, as the page's check chips do. A correct solution makes both sides equal in both equations; a solution that satisfies only one equation is wrong.
Results are estimates only and do not constitute professional advice.