How to solve a system of equations

A system of equations is a set of two or more equations that share the same unknowns. Instead of solving each equation in isolation, you solve them together to find values of the variables that make all equations true at the same time. Systems show up everywhere: budgeting (income vs expenses), business (supply vs demand), chemistry (balancing mixtures), physics (forces), and of course math classes.

This page is a fast, mobile-friendly System of Equations Solver that focuses on the most common case: two linear equations with two unknowns (usually written as x and y). You can enter the equations in coefficient form (recommended for accuracy), and the calculator returns:

• The solution (x and y), or a clear message if there is no unique solution.
• The determinant and what it means.
• Step-by-step elimination (so you can learn, not just copy).
• Verification by plugging the solution back into both equations.

• One unique solution: the lines cross at exactly one point. This is the “normal” case.
• No solution: the lines are parallel (same slope, different intercept). The equations contradict.
• Infinitely many solutions: the lines are the same line (equations are equivalent).

In linear algebra language, you can write a 2×2 system as: a·x + b·y = e and c·x + d·y = f. The key number is the determinant: Δ = a·d − b·c. If Δ ≠ 0, the system has a unique solution. If Δ = 0, the system is either inconsistent (no solution) or dependent (infinitely many solutions).

• Screenshot-ready: the result box includes the final answer + a clean step summary.
• Study hack: paste problems from homework, practice tests, or tutoring sessions.
• Real-world modeling: compare two plans, rates, mixtures, or constraints instantly.
• Share buttons: send the solved system to friends/classmates in one tap.

Elimination means combining equations to “eliminate” one variable, leaving a single-variable equation you can solve. For example, if you have:

• Equation (1): a·x + b·y = e
• Equation (2): c·x + d·y = f

You pick a variable to eliminate (x or y). Suppose you eliminate x. Multiply equation (1) by c and equation (2) by a so both have the same x coefficient:

• (1)×c: (a·c)x + (b·c)y = e·c
• (2)×a: (c·a)x + (d·a)y = f·a

Now subtract one from the other to cancel x: (b·c − d·a)·y = (e·c − f·a). Solve for y, then plug y back into either original equation to get x.

Cramer’s Rule gives direct formulas using determinants (perfect for calculators). Using Δ = a·d − b·c:

• x = Δx / Δ, where Δx = e·d − b·f
• y = Δy / Δ, where Δy = a·f − e·c

If Δ ≠ 0, you’re done. If Δ = 0, Cramer’s Rule tells you there is no unique solution, so you switch to reasoning about parallel vs identical lines (or use elimination to check consistency).

Example 1 (unique solution)

Solve: 2x + 3y = 13 and x − y = 1. Here a=2, b=3, e=13, c=1, d=−1, f=1. Compute Δ = a·d − b·c = 2·(−1) − 3·1 = −2 − 3 = −5. Since Δ ≠ 0, there is a unique solution. Now Δx = e·d − b·f = 13·(−1) − 3·1 = −13 − 3 = −16, so x = (−16)/(−5) = 3.2. And Δy = a·f − e·c = 2·1 − 13·1 = −11, so y = (−11)/(−5) = 2.2. Check quickly: 2(3.2)+3(2.2)=6.4+6.6=13 ✓ and 3.2−2.2=1 ✓.

Example 2 (no solution)

Solve: 2x + 4y = 10 and x + 2y = 8. Multiply the second equation by 2: 2x + 4y = 16. But the first says 2x + 4y = 10. Same left side, different right side → contradiction → no solution (parallel lines).

Example 3 (infinitely many solutions)

Solve: 3x − 6y = 9 and x − 2y = 3. Multiply the second equation by 3: 3x − 6y = 9, which matches the first exactly. So the equations are the same line, meaning infinitely many solutions. You can describe solutions as: x − 2y = 3 → x = 3 + 2y (pick any y and compute x).

When you press Solve, the calculator:

1. Reads your coefficients (a, b, e, c, d, f).
2. Computes the determinant Δ = a·d − b·c.
3. If Δ ≠ 0, computes x and y using Cramer’s Rule and formats the answer neatly (fractions when possible).
4. If Δ = 0, checks whether the system is inconsistent or dependent by comparing ratios.
5. Generates elimination steps you can copy into homework solutions.
6. Verifies the result by plugging x and y back into both equations (so you can trust it).

• Use integers when possible. Decimals are fine, but can create rounding noise.
• If your equation is like “x + y = 5”, enter a=1, b=1, e=5.
• If your equation is like “−y = 7”, enter a=0, b=−1, e=7.
• For fractions (like 1/2), type 1/2 or 0.5.

• Sign errors: Watch negatives. “x − y = 1” means b = −1.
• Moved terms: Always rewrite to a·x + b·y = e before entering.
• Swapped coefficients: Make sure the x coefficient goes in the x box, y coefficient in y box.
• Parallel lines confusion: If Δ = 0, the system is not “unsolvable”; it just doesn’t have a unique point.

Want to go beyond 2 variables? The same ideas scale up: systems become matrices, determinants still matter, and elimination becomes Gaussian elimination. But for most homework, business modeling, and quick checks, the 2×2 solver is the sweet spot: fast, clear, and easy to verify.

Frequently Asked Questions

• What does “system of equations” mean?

  It’s a set of equations that share the same variables. You’re looking for values (like x and y) that make every equation true at the same time.

• What kinds of systems does this solver handle?

  This calculator solves two linear equations with two unknowns in coefficient form: a·x + b·y = e and c·x + d·y = f. It also detects when the system has no solution or infinitely many solutions.

• Can I type fractions like 1/3?

  Yes. You can enter fractions (e.g., 1/3) or decimals (e.g., 0.3333). The solver parses both. For clean-looking results, fractions often keep things exact.

• Why does the determinant matter?

  The determinant Δ = a·d − b·c tells you whether the two lines are “tilted differently.” If Δ ≠ 0, the lines intersect once → one unique solution. If Δ = 0, the lines are parallel or the same line → no unique solution.

• What does “no solution” look like in real life?

  It means your constraints contradict each other. For example, one equation says the total must be 10, but another equation (with the same left side) says the total must be 16. Both can’t be true.

• What does “infinitely many solutions” mean?

  The equations are actually the same relationship written two different ways. Instead of one point, you get a whole line of solutions. You can describe it with a parameter like y = t.

• How accurate is the answer?

  If you use integers or fractions, the result is exact. With decimals, your answer may be rounded. The solver also verifies the solution by plugging it back into both equations.

• How do I check the solution quickly?

  Substitute x and y into each equation and make sure the left side equals the right side. The calculator prints those checks automatically under “Verification.”

• Can I use this for word problems?

  Yes—just translate the story into two linear equations. Common patterns include: mixture problems, rate/time problems, and comparing two pricing plans.

• Does this work offline?

  Yep. Everything runs in your browser with plain JavaScript. No login, no API, no tracking needed for calculations.

Formula breakdown (Cramer’s Rule)

For the system: a·x + b·y = e and c·x + d·y = f, the determinant is Δ = a·d − b·c. If Δ ≠ 0, you have a unique solution:

• x = (e·d − b·f) / Δ
• y = (a·f − e·c) / Δ

The reason this works is that determinants measure how “non-parallel” the equations are. If the two equations point in different directions, they meet once (unique solution). If they point in the same direction, they never meet (no solution) or overlap (infinitely many solutions).

• If Δ ≠ 0, it returns x and y immediately (unique solution).
• If Δ = 0 and the equations are proportional but constants are not, it returns no solution.
• If Δ = 0 and everything is proportional (including constants), it returns infinitely many solutions.

Before you submit your homework

• Rewrite equations in standard form: a·x + b·y = e.
• Make sure you copied signs correctly (minus signs cause most mistakes).
• Check Δ = a·d − b·c: if Δ = 0, explain “no solution” vs “infinite solutions.”
• Show substitution verification (plug x and y back in).
• Use exact fractions if your instructor prefers exact answers.

Screenshot tip: After solving, take a screenshot of the result + steps block. It’s clean enough for notes and tutoring chats.