A system of linear equations has one solution when the lines (or planes) cross at a single point. 2x + 3y = 13 and x − y = −1 meet at x = 2, y = 3. Type each equation the way you write it on paper, with terms on either side of the equals sign; the solver puts them in standard form, works out the determinants, gives each unknown as an exact fraction when it can and checks the answer in the first equation.
The steps, on the example
- Standard form: 2x + 3y = 13 and x − y = −1.
- D = 2 × (−1) − 3 × 1 = −5.
- Dx: replace the x column by 13 and −1: 13 × (−1) − 3 × (−1) = −10, so x = −10 ÷ −5 = 2.
- Dy: replace the y column: 2 × (−1) − 13 × 1 = −15, so y = −15 ÷ −5 = 3.
- Check: 2 × 2 + 3 × 3 = 13.
The same method extends to three equations with three unknowns, using 3 × 3 determinants. Choose “Three equations” and add the third line.
One, none or infinitely many
| Example | D | What it means |
|---|---|---|
| 2x + 3y = 13, x − y = −1 | −5 | One solution: x = 2, y = 3 |
| 3x + 2y = 7, x − 4y = 0 | −14 | One solution: x = 2, y = 1/2 |
| x + y = 2, 2x + 2y = 5 | 0 | No solution: parallel lines |
| x + y = 2, 2x + 2y = 4 | 0 | Infinitely many: the same line |
Turning a word problem into a system
“Two adult tickets and three child tickets cost 13; one adult ticket costs 1 more than a child ticket.” Call the prices x and y: 2x + 3y = 13 and x − y = 1. The solver gives x = 3.2 and y = 2.2. Name the unknowns first, write one equation per fact, and check that the answer makes sense (prices cannot be negative).
Homework does not need a connection: the solver works offline once zet4u is saved on your device.
Frequently asked questions
How do I solve two equations with two unknowns?
Eliminate one unknown. For 2x + 3y = 13 and x − y = −1: multiply the second by 3 and add, giving 5x = 10, so x = 2 and y = 3. Cramer's rule gives the same answer with determinants.
What is Cramer's rule?
x = Dx ÷ D, y = Dy ÷ D, where D is the determinant of the coefficients and Dx, Dy are the determinants with the x or y column replaced by the numbers on the right of the equals sign.
What if the system has no solution?
Then D = 0 and the equations contradict each other: x + y = 2 and 2x + 2y = 5 are parallel lines that never meet. If one equation is just a multiple of the other, as with x + y = 2 and 2x + 2y = 4, there are infinitely many solutions.
Can I type fractions and decimals?
Yes: 0.5x, x/2 and 1/2x all work, and so do terms on both sides, such as 3x = y + 7. Use x, y and z as the unknowns.
Why are some answers shown as fractions?
When every coefficient is a whole number, the answer is an exact fraction such as 7/3, with the decimal next to it. Rounding 2.333… would lose the exact value.
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