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 to 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 →

Systems of Equations With 2 to 3 Unknowns

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