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System of Equations Solver: Two Linear Equations

This free solver finds x and y for two linear equations in the form ax + by = c. It uses Cramer’s rule, shows the determinants and tells you when the lines are parallel or the same.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

Two linear equations describe two straight lines, and their solution is the point where they cross. Mixing problems, cost comparisons and many school problems lead to such a pair.

Enter the coefficients of both equations.

How to Use This Tool

  1. Enter the value or values asked for in the boxes.
  2. Choose the unit label if there is a choice.
  3. Press Calculate.
  4. Read every result in the table and copy what you need.

How the Calculation Works

The determinant is D = a1 b2 − a2 b1. If D is not zero, x = (c1 b2 − c2 b1) ÷ D and y = (a1 c2 − a2 c1) ÷ D. If D is zero, the system has no solution or infinitely many, which the tool tells apart by checking the ratios of the coefficients.

Worked Example

For 2x + y = 5 and x − y = 1, D = −3, x = 2 and y = 1. For x + y = 1 and 2x + 2y = 3, the lines are parallel and there is no solution.

Common Uses

  • Algebra homework.
  • Mixing and cost problems.
  • Finding where two lines meet.
  • Checking hand work.

Tips and Common Mistakes

  • Write each equation in the form ax + by = c first.
  • Check the answer in both equations.
  • Use 0 for a missing term.
  • Measure carefully: a small error in a length is squared in an area and cubed in a volume, so a 5 percent error in a side can become a 10 percent error in the area or a 15 percent error in the volume.
  • Round only at the end. If you copy a result into another calculation, copy the full number from the tool rather than a rounded one, so small errors do not build up.

Next, you may need the decimal to fraction converter, simplify radical calculator, fraction to decimal converter, rhombus calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

A way to solve a system with determinants. For two equations, x = (c1b2 − c2b1) ÷ D and y = (a1c2 − a2c1) ÷ D, where D = a1b2 − a2b1.
Then the lines are parallel or the same. If they are the same line there are infinitely many solutions, otherwise there is none.
a = 2, b = 1 and c = 5. A missing term has a coefficient of 0.
Enter them as decimals, for example 0.5 for one half.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.