System of Equations Calculator
Solve a system of two linear equations in two variables (a₁x + b₁y = c₁, a₂x + b₂y = c₂) using Cramer's rule, with a diagnosis of no or infinite solutions.
Comment utiliser cette calculatrice
- 1Write each equation in the form ax + by = c.
- 2Enter the coefficients a, b, and the constant c for the first equation.
- 3Do the same for the second equation.
- 4Read the solution (x, y), or a diagnosis of no or infinitely many solutions.
Comment ça marche
Two linear equations (Cramer's rule)
a₁x + b₁y = c₁, a₂x + b₂y = c₂ D = a₁b₂ − a₂b₁ x = (c₁b₂ − c₂b₁) ÷ D y = (a₁c₂ − a₂c₁) ÷ D
A system of two linear equations in two unknowns asks for the values of x and y that satisfy both equations at once — geometrically, the point where two lines cross. Cramer's rule solves it elegantly using determinants. The main determinant D is formed from the coefficients of x and y; if it is non-zero, the system has exactly one solution, given by two more determinants divided by D. The x-solution uses a determinant in which the x-coefficients are replaced by the constants, and the y-solution replaces the y-coefficients instead. When D equals zero, the two lines are parallel and never meet (no solution) or are actually the same line (infinitely many solutions), which the values of the other determinants distinguish. Cramer's rule is a clean, formula-driven alternative to substitution or elimination, and it generalises to larger systems, though for two variables all three methods reach the same answer.
Exemple détaillé
For x + y = 5 and 2x − y = 4, the main determinant is D = (1)(−1) − (2)(1) = −3. Then x = ((5)(−1) − (4)(1)) ÷ −3 = −9 ÷ −3 = 3, and y = ((1)(4) − (2)(5)) ÷ −3 = −6 ÷ −3 = 2. The lines cross at (3, 2), which satisfies both equations.
System of Equations Calculator : le guide complet
Where two lines meet
A system of two linear equations is, at heart, a geometric question: each equation describes a straight line in the plane, and solving the system means finding the point (or points) that lie on both lines at once. Most of the time, two lines cross at exactly one point, and the coordinates of that intersection are the unique solution — the single pair of x and y values that make both equations true simultaneously. This geometric picture is the key to understanding not just how to solve the system but why the different outcomes arise.
The intersection view also explains the special cases. If the two lines happen to be parallel, they never meet, and the system has no solution — there is no pair of values that satisfies both. If the two equations describe the very same line, then every point on that line satisfies both, giving infinitely many solutions. These three possibilities — one solution, none, or infinitely many — are the only outcomes for a system of two linear equations, and which one you get depends entirely on how the two lines are oriented relative to each other.
Cramer's rule and the determinant
Cramer's rule provides a direct formula for the solution using determinants, quantities computed from the coefficients that measure something important about the system. The main determinant, D, is calculated from the coefficients of x and y as a₁b₂ − a₂b₁. Its value acts as a diagnostic. When D is non-zero, the two lines have different slopes, they must cross somewhere, and there is exactly one solution, which Cramer's rule delivers by dividing two further determinants by D. The method is systematic and mechanical, which makes it easy to apply and to program.
When D equals zero, something has gone wrong with the neat single-solution case: the lines are parallel or identical, because a zero main determinant means the coefficient rows are proportional and the slopes match. At that point Cramer's rule cannot divide, and the system either has no solution or infinitely many. Examining the other determinants, or simply checking whether the equations are consistent multiples of one another, tells the two apart. The determinant thus does double duty — it both computes the answer when one exists and flags when no unique answer does. This is a small preview of a much deeper role determinants play throughout linear algebra, where the vanishing of a determinant always signals a loss of the clean, invertible behaviour that a non-zero one guarantees.
Solving systems in the real world
Systems of linear equations appear whenever two quantities are linked by two separate conditions, which happens constantly in practical problems. A classic example is mixing: combining two solutions of known concentrations to reach a target volume and concentration gives two equations in the two unknown amounts. Break-even and supply-and-demand problems in economics find the point where two linear relationships coincide. Puzzles about the cost of two items given two total purchases, or the speeds of a boat with and against a current, all reduce to two equations in two unknowns.
While a two-variable system can be solved by substitution, elimination, or Cramer's rule interchangeably, the real importance of these methods is that they scale. Larger systems — three, ten, or thousands of equations in as many unknowns — arise throughout engineering, physics, economics, computer graphics, and machine learning, and they are solved by systematic generalisations of exactly these ideas, especially the determinant and matrix methods that Cramer's rule foreshadows. Learning to solve a two-by-two system cleanly, and to read the determinant as a signal of how many solutions exist, is therefore the entry point to linear algebra, one of the most widely applied branches of mathematics. The humble pair of lines crossing on a graph is the first step toward solving the enormous systems that model the modern world.
Questions fréquentes
How do I solve a system of two equations?
Write both as ax + by = c, then use Cramer's rule: D = a₁b₂ − a₂b₁, x = (c₁b₂ − c₂b₁) ÷ D, y = (a₁c₂ − a₂c₁) ÷ D. For x + y = 5 and 2x − y = 4, this gives x = 3, y = 2. Substitution and elimination reach the same answer.
What does the determinant tell me?
The main determinant D diagnoses the system. If D isn't zero, there's exactly one solution — the lines cross at a point. If D is zero, the lines are parallel (no solution) or identical (infinitely many solutions). A non-zero determinant guarantees a unique answer.
What if there's no solution?
The two lines are parallel — same slope, different intercepts — so they never meet, and no pair of values satisfies both equations. This shows up as a zero main determinant with a non-zero solution determinant. Geometrically, the lines run alongside each other forever.
When are there infinitely many solutions?
When both equations describe the same line — one is a multiple of the other. Every point on the line satisfies both, so there are infinitely many solutions. This happens when all the determinants are zero, meaning the equations are truly equivalent.