Two-variable linear system
Solve two independent linear equations for two unknowns.
Two-variable linear system is one of 1 systems formula in the algebra section of this library, and it is used at high school level.
Why two-variable linear system works
Eliminating one variable from the pair by hand leaves exactly the denominator shown here multiplying the surviving variable. That denominator measures how differently the two lines are tilted, and each numerator is the same combination with the constants substituted into the column of the variable being solved for.
What each symbol means
The $a_i,b_i$ are coefficients and $c_i$ constants.
Two-variable linear system: when it holds
The determinant $a_1b_2-b_1a_2$ must be nonzero for one unique solution.
When it stops applying
When the denominator is zero these formulas give nothing, because the two lines are parallel or identical. For 2x + 4y = 6 and x + 2y = 3 the denominator is 4 minus 4, and the two equations describe the same line, so there are infinitely many solutions.
Two-variable linear system: a worked example
For $x+y=5$ and $x-y=1$, the formulas give $x=3,y=2$.
The mistake to avoid
What people do: Adding the two cross products in the denominator instead of subtracting them.
Why it goes wrong: For x + y = 5 and x - y = 1, subtracting gives -2 and the answers x = 3 and y = 2. Adding gives 0, which pretends the system has no unique solution when it plainly does.
Do this instead: Multiply along the main diagonal, take away the product along the other diagonal, and compute that single number once before touching either numerator.
Two-variable linear system: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. The $a_i,b_i$ are coefficients and $c_i$ constants.
- Check the conditions before substituting. The determinant $a_1b_2-b_1a_2$ must be nonzero for one unique solution.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Systems
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where two-variable linear system comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Systems of Equations — the lesson behind this formula: use substitution, elimination, and graphs to find shared solutions.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about two-variable linear system
Do both equations have to be arranged the same way?
Yes. Line the x terms above each other, the y terms above each other, and the constants on the right, or the coefficients you pull out will belong to the wrong slots.
What do I use when one equation is missing a variable?
Use a coefficient of zero for it. An equation like 3x = 12 is really 3x + 0y = 12, and the formulas handle it without any special treatment.
Is substitution or elimination faster?
Usually yes for a single system done by hand, especially when one coefficient is 1. These formulas earn their keep when you must solve many systems with the same shape or program the solution.
How do I tell no solution from infinitely many when the denominator is zero?
Check whether one whole equation is a multiple of the other. If it is, they are the same line with infinitely many shared points; if not, they are parallel and share none.