What's the difference between substitution and elimination?

Substitution solves one equation for a variable and plugs it into the others; elimination adds or subtracts scaled equations to cancel a variable. Both reach the same solution — pick whichever keeps the arithmetic simplest.

When does a system have no solution or infinitely many?

If the equations are inconsistent (parallel lines/planes) there is no solution; if they are dependent (the same line/plane) there are infinitely many. A unique solution exists only when the coefficient determinant is non-zero.

Systems of Equations With 2–3 Unknowns (Step by Step)

Q: How do I solve a system on the calculator?

Switch to EQN mode, choose simultaneous equations in 2 or 3 unknowns, and enter each equation's coefficients and constant. The calculator returns x, y (and z) directly.

A system of equations asks for the values that satisfy several equations at once. With a simultaneous equation solver you can confirm the answer instantly, but it helps to know the two by-hand methods first.

Substitution (2 unknowns)

Solve the second equation for x: x = y + 1. Substitute into the first:

2(y + 1) + y = 5  →  3y + 2 = 5  →  y = 1,  then x = 2

Elimination (2 unknowns)

Add the two equations to cancel y:

(2x + y) + (x − y) = 5 + 1  →  3x = 6  →  x = 2,  then y = 1

Either way, the solution is x = 2, y = 1. ✔

Three unknowns

The same ideas scale up — eliminate one variable to drop to a 2×2 system, then back-substitute.

x + y + z = 6
2x − y + z = 3
x + 2y − z = 2

Solving gives x = 1, y = 2, z = 3. Check the second equation: 2 − 2 + 3 = 3 ✔.

The EQN solver

Switch to EQN mode, choose simultaneous equations in 2 or 3 unknowns, and enter each equation's coefficients and constant:

(2, 1, 5)
(1, −1, 1)   →   x = 2, y = 1

For the matrix viewpoint of the same problem, see solving simultaneous equations with matrices; for the whole solver, the equation solver pillar guide.

No solution / infinitely many

Watch for these

If two equations describe parallel lines or planes, the system is inconsistent — no solution. If they describe the same line or plane, they're dependent — infinitely many solutions. A unique solution exists only when the coefficient determinant is non-zero (see 3×3 determinants).

Frequently asked questions

Substitution vs elimination?

Substitution plugs one variable into the others; elimination cancels a variable by adding/subtracting. Same answer — pick the simpler arithmetic.

How do I solve a system on the calculator?

EQN mode → simultaneous (2 or 3 unknowns) → enter coefficients and constants → read x, y, z.

When is there no/infinite solutions?

Inconsistent equations → none; dependent equations → infinitely many; unique only if the determinant ≠ 0.

Solve your system

Enter the coefficient rows in EQN mode and read x, y and z at once.

Open the simultaneous equation solver →