System of Equations Calculator

Q: How do I enter an equation like y = 3x − 2?

Rearrange it so the variables are on the left: −3 x + y = −2. Then enter −3, 1 and −2.

Solve 2 or 3 linear equations at once. Get the solution as exact fractions and decimals, the full working, a check, and a graph of where the lines meet.

What is a system of equations?

A system of equations is a set of equations that share the same unknowns. Solving it means finding values that make every equation true at the same time. For two equations in x and y, each equation is a straight line, and the solution is the point where the lines cross.

One solution, no solution or infinitely many

Type	What the lines do	Example
One solution	Cross at one point	2x + y = 5 and x − y = 1 meet at (2, 1)
No solution	Parallel, never meet	x + 2y = 3 and 2x + 4y = 7
Infinitely many	The same line	x + 2y = 3 and 2x + 4y = 6

Methods for solving

Elimination. Add or subtract the equations to cancel one variable. For 2x + y = 5 and x − y = 1, adding them cancels y: 3x = 6, so x = 2. Substituting back gives y = 1.

Substitution. Solve one equation for a variable and plug it into the other. From x − y = 1, x = y + 1; then 2(y + 1) + y = 5 gives y = 1 and x = 2.

Graphing. Draw both lines and read off where they cross. It shows the idea clearly but is only exact when the answer lands on grid points.

Cramer’s rule. Uses determinants: x = D_x ÷ D and y = D_y ÷ D. It is quick for 2 and 3 equations, keeps fractions exact, and immediately tells you when there is no unique solution (D = 0). The calculator above shows the working this way.

Worked example: a word problem

A school play sold 30 tickets for $300. Adult tickets cost $12 and child tickets cost $8. How many of each were sold?

Let a be adult tickets and c child tickets. Then a + c = 30 and 12a + 8c = 300. Substituting c = 30 − a gives 12a + 240 − 8a = 300, so 4a = 60 and a = 15. That leaves c = 15: 15 adult and 15 child tickets. Try it with the “Ticket problem” example above.

Three equations

With three unknowns, each equation is a plane in 3D space, and a unique solution is the single point where all three planes meet. The same methods extend: eliminate one variable to get two equations in two unknowns, or use Cramer’s rule with 3 × 3 determinants. For example, x + y + z = 6, 2y + 5z = −4 and 2x + 5y − z = 27 has the solution x = 5, y = 3, z = −2.

Frequently asked questions

Can it solve nonlinear systems?

No, this calculator handles linear equations only, where each variable appears to the first power. For a single quadratic equation, use the quadratic formula calculator.

How do I enter an equation like y = 3x − 2?

Rearrange it so the variables are on the left: −3x + y = −2. Then enter −3, 1 and −2.

Can I share a system with someone?

Yes. After you press Solve, the page address holds your numbers, so the link opens the same system.

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Last updated September 27, 2026.