A system of equations is a set of equations that must all be true at the same time. Solving it means finding the values that satisfy every equation simultaneously — geometrically, the point where the graphs meet. The methods in this guide differ in bookkeeping, not in mathematics: substitution, elimination and matrices all enforce the same requirement, and all three can return the same three verdicts. What changes is which method keeps the arithmetic cleanest for the system in front of you, and recognising that choice is most of the skill.

What a system is: two conditions, one pair of values

One equation in two unknowns, such as x + y = 10, has infinitely many solutions: every pair that adds to 10. A second equation, such as x − y = 2, is a second condition, and the pair must satisfy both. Adding the conditions narrows the possibilities, usually to a single pair — here x = 6, y = 4. The system is the pair of conditions together, and the solution is the pair that survives both. That is why a system is solved as a whole: solving one equation alone tells you nothing about the system, and a value that satisfies one equation but not the other is not a solution at all.

Geometrically, each linear equation in two unknowns is a straight line, and the solution is the point the lines share. Two lines in a plane either cross once, never meet, or coincide — and those are exactly the three verdicts. The graph is a useful check even when the algebra is the working method: if the algebra says “no solution”, the lines should look parallel, and if it says “infinitely many”, the two equations should turn out to be multiples of each other.

Substitution: follow the variable that is already alone

Substitution works by replacing a variable with its equal. If one equation gives x = 2y + 3, substitute 2y + 3 for x everywhere it appears in the other equation. The system collapses to one equation in y, which solves directly; the value of y then goes back into the isolated equation to give x. For the system x = 2y + 3 with 3x + 2y = 19, substitution gives 3(2y + 3) + 2y = 19, so 8y = 10 and y = 1.25, and x = 2(1.25) + 3 = 5.5. Check: 3(5.5) + 2(1.25) = 16.5 + 2.5 = 19. The check is part of the method, not an optional extra — substitution creates many opportunities for a sign or distribution slip, and the original equations catch them all.

Substitution is the natural method when an equation already isolates a variable, when a coefficient is 1 (so isolating costs one step), or when one equation is much simpler than the other. It is also the method that generalises to non-linear systems, where elimination’s cancellations are unavailable: solve one equation, substitute, and solve what remains, checking every root in both originals.

Elimination: make a variable cancel, then read the rest

Elimination adds or subtracts whole equations to remove a variable. For 2x + 3y = 12 and 2x − y = 4, subtracting the second from the first removes x entirely: 4y = 8, so y = 2, and back-substitution gives 2x − 2 = 4, so x = 3. When the coefficients do not already match, multiply one equation (or both) by a number that makes them match — the equation’s solution set is unchanged by multiplying through by a non-zero constant. The art is choosing the multiplier that keeps the numbers small: for 3x + 2y = 7 and 5x − 3y = 1, multiplying the first by 3 and the second by 2 makes the y-coefficients +6 and −6, and adding gives 19x = 23.

Elimination shows the three verdicts most plainly. If the variables cancel and a true statement remains (0 = 0), the equations describe the same line. If a false statement remains (0 = 5), the lines are parallel. If one variable survives, the system has the single solution that variable and its partner give. A common slip is to multiply one side of an equation and forget the other; multiplying an equation means multiplying every term, including the constant.

Matrices: the same elimination, kept in rows

A matrix writes a system as a rectangle of coefficients with the constants alongside — the augmented matrix — and Gaussian elimination performs the same operations as elimination, tracked as row operations: swap two rows, multiply a row by a non-zero number, or add a multiple of one row to another. Each operation preserves the solution set, and the goal is a triangular form where the last row involves one variable, the row above involves two, and so on, so the values fall out by back-substitution. The calculator on this page displays those steps in the open, which is the best way to learn the method: the steps are elimination, without the variables written out each time.

Matrices earn their keep at three or more unknowns, where keeping track of three equations by hand invites exactly the bookkeeping errors rows prevent. Cramer’s rule — each unknown as a ratio of determinants — is a second matrix method, elegant for two or three unknowns and impractical beyond them, because determinants grow expensive to compute. For larger systems, elimination in matrix form (or its computerised equivalent) is the standard tool, and the verdicts read off the same way: a row of zeros with a non-zero constant means no solution, and a row of zeros throughout means a free variable and infinitely many solutions.

Simultaneous equations and linear equations: the same idea, two names

“Simultaneous equations” is the name most school systems use for a system that must hold at the same time, and “system of linear equations” is the same object named by its form. A linear equation in two unknowns graphs as a straight line; a system of them is a set of lines asking for their common point. The vocabulary matters only because search and textbooks use both: a question phrased as “solve the simultaneous equations” and one phrased as “solve the system” want the same working, and the same three verdicts.

Linear systems also appear disguised. Two quantities that grow at different rates and meet at a break-even point, two mixtures that must combine to a target strength, or a journey split into two legs with a known total time and distance, are all systems once the unknowns are named. The disguise is the point of word problems: the mathematics is the same elimination or substitution, and the difficulty is entirely in the translation from sentences to equations.

Word problems: the setup is the solution

Name the unknowns with units before writing anything: let x be the number of adult tickets (tickets), let y be the number of child tickets (tickets). Then translate each sentence into exactly one equation, keeping the units honest — tickets plus tickets, dollars plus dollars, never tickets plus dollars. Two unknowns need two independent equations; a second sentence that merely restates the first (in different words) is not a second equation, and the system will return infinitely many solutions to tell you so.

Work an example. A theatre sells 120 tickets for $480. Adult tickets cost $5, child tickets $3. Let a be adult tickets and c child tickets: a + c = 120 and 5a + 3c = 480. Multiply the first by 3: 3a + 3c = 360. Subtract from the second: 2a = 120, so a = 60 and c = 60. Check in the originals: 60 + 60 = 120, and 5(60) + 3(60) = 300 + 180 = 480. The check is not ceremony — it is the step that catches a misread price or a swapped variable, which are the commonest word-problem errors and the ones no method can correct on its own.

Mistakes: signs, distribution, and the unchecked answer

Sign errors dominate. Subtracting an equation subtracts every term, including a negative coefficient that becomes positive; distributing a multiplier across a bracket distributes to every term, including the constant. Both errors produce a plausible-looking wrong answer that only the check in the original equations exposes. Fractions deserve the same care: keep them as fractions through the working — the solver on this page never rounds mid-solve, for exactly this reason — and convert to decimals only at the end, if at all.

The second family of mistakes is conceptual: treating “no solution” as a failure and forcing an answer, treating “infinitely many” as one answer picked at random, or reporting a solution that satisfies one equation only. The verdict is the answer when the verdict is what the system gives, and the general solution (for example, “any pair with y = 2x + 1”) is the correct way to report infinitely many solutions, not a single pair chosen for convenience.

Frequently asked questions

What is a system of equations?

A set of two or more equations that must all be true at the same time, sharing the same unknowns. The solution is the set of values that satisfies every equation simultaneously — geometrically, the point or points where the graphs meet. A system of two linear equations in two unknowns has one solution, no solution, or infinitely many solutions, depending on whether the lines cross, are parallel, or coincide.

How does the elimination method work?

Elimination adds or subtracts whole equations — after multiplying an equation through by a number if needed — so that one variable cancels. What remains is one equation in one unknown, which solves directly, and the value is substituted back to find the other unknown. If the variables cancel and leave a true statement such as 0 = 0, the system has infinitely many solutions; if they leave a false statement such as 0 = 5, it has none.

What does it mean when a system has no solution?

It means no pair of values satisfies both equations at once. For two linear equations, the lines are parallel: same slope, different intercept, never meeting. The system is inconsistent, and the correct answer is to say so — there is no pair to find, and forcing one by ignoring an equation is not solving the system. In a word problem, no solution usually means the stated conditions contradict each other, and the conditions, not the method, need revisiting.

What does it mean when a system has infinitely many solutions?

It means the equations describe the same line (or the same plane, in three unknowns), so every point on it satisfies the system. Elimination reveals this when the variables cancel and leave a true statement such as 0 = 0. The answer is reported as a general solution — for example, all pairs with y = 2x + 1 — not as a single pair. In a word problem it usually means one of the conditions was not independent, and a further condition is needed to pin the answer down.

How do you solve a system with three unknowns?

By the same elimination, extended: use two pairs of equations to eliminate one variable, leaving a two-unknown system, then solve that as usual and back-substitute. Matrices make the bookkeeping manageable — Gaussian elimination on the augmented matrix performs the same steps as row operations. The verdicts are the same: one point where three planes meet, no common point, or a line or plane of solutions when the equations overlap.

Should I use substitution or elimination?

Use substitution when one equation already isolates a variable or has a coefficient of 1, so isolating costs one step. Use elimination when the coefficients line up, or can be made to line up with one small multiplier, so a variable cancels in one addition. Use matrices for three or more unknowns, where row bookkeeping prevents errors. All three give the same solution to the same system; the choice is about keeping the arithmetic clean, not about correctness.