Simultaneous Equation Calculator — Solve 2-Variable Systems
Solve systems of two linear equations with two unknowns. Enter the 6 coefficients and get x, y values with full step-by-step solution using the elimination method.
Enter coefficients for the system: ax + by = c and dx + ey = f
Equation 1: x + y =
Equation 2: x + y =
Examples:
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y
Step-by-Step Solution
How to Solve Simultaneous Equations
Given the system ax + by = c and dx + ey = f, the solution uses Cramer's Rule based on determinants:
- Determinant D = ae - bd
- x = (ce - bf) / D
- y = (af - cd) / D
If D = 0, the system either has no solution (parallel lines) or infinitely many solutions (same line).
Frequently Asked Questions
What is a system of simultaneous equations?
A system of simultaneous equations is a set of two or more equations with the same unknowns (variables) that must all be satisfied at the same time. The solution is the values of the unknowns that satisfy every equation in the system simultaneously.
What is the elimination method?
The elimination method involves multiplying one or both equations by constants so that when you add or subtract them, one variable is eliminated. You then solve for the remaining variable and substitute back to find the other.
What is the substitution method?
The substitution method involves solving one equation for one variable, then substituting that expression into the other equation. This gives you one equation with one unknown, which you can solve directly.
When does a system have no solution or infinite solutions?
A system has no solution (inconsistent) when the equations represent parallel lines — same slope, different y-intercepts. It has infinite solutions (dependent) when both equations represent the same line. This happens when the determinant of the coefficient matrix equals zero.
What is Cramer's Rule?
Cramer's Rule uses determinants to solve a system. For ax + by = c and dx + ey = f, the determinant D = ae - bd. Then x = (ce - bf) / D and y = (af - cd) / D. This calculator uses this method internally.