Simultaneous Equations Calculator

A system of equations is a set of two or more equations that must all be satisfied at the same time, and solving it means finding the values of the unknowns that make every equation true simultaneously. Two-variable linear systems of the form ax plus by equals c arise constantly in maths courses and practical problems: finding where two lines intersect on a graph, working backwards from combined totals to find individual prices, solving mixture problems, balancing supply and demand, and many engineering and physics problems. This calculator solves a system of two linear equations in two variables using Cramer's rule, a determinant-based method that expresses each unknown directly as a ratio of two values computed from the coefficients. You enter the coefficient of x, the coefficient of y, and the constant on the right-hand side for each of the two equations, and the calculator returns the values of x and y, the main determinant of the coefficient matrix, and the step-by-step working so you can follow exactly how the answer was reached. Negative coefficients work fine; simply enter a negative number. If the main determinant is zero, the equations either contradict each other or are equivalent to the same line, and no unique solution exists. The calculator detects this and tells you rather than returning a meaningless result. Enter your coefficients to see the solution update as you type.

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Equation 1: ax + by = c

x + y =

Equation 2: dx + ey = f

x + y =

x = 3, y = 2

solution

Determinant (D)-5

Dx / Dy-15 / -10

Working: D = (2)(-1) - (1)(3) = -5. Dx = (12)(-1) - (1)(3) = -15. Dy = (2)(1) - (1)(12) = -10. x = -15/-5 = 3. y = -10/-5 = 2.

How it works

For the system ax + by = c and dx + ey = f, Cramer's rule computes the main determinant D = ae minus bd. The determinant Dx is found by replacing the x-column with the constants: ce minus bf. The determinant Dy is found by replacing the y-column: af minus dc. The solutions are x = Dx divided by D and y = Dy divided by D. When D equals zero, no unique solution exists and the calculator reports this clearly.

Worked example

Using the defaults: 2x + 3y = 12 and x minus y = 1. The main determinant D = (2)(minus 1) minus (1)(3) = minus 5. Dx = (12)(minus 1) minus (1)(3) = minus 15. Dy = (2)(1) minus (1)(12) = minus 10. Therefore x = minus 15 divided by minus 5 = 3 and y = minus 10 divided by minus 5 = 2. Check: 2(3) + 3(2) = 12 and 3 minus 2 = 1. Both equations are satisfied.

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Frequently asked questions

What is Cramer's rule?

Cramer's rule solves a linear system by expressing each unknown as a ratio of two determinants. The denominator is the determinant of the coefficient matrix. The numerator for each variable is the determinant of the same matrix but with that variable's column replaced by the constant terms. Dividing gives the value of the unknown.

What happens if the determinant is zero?

If the main determinant is zero, the system has no unique solution. The two equations are either inconsistent (parallel lines with no intersection) or dependent (the same line written differently, giving infinitely many solutions). The calculator reports this instead of dividing by zero.

How do I enter the equations?

Enter the coefficient of x, the coefficient of y, and the constant on the right-hand side for each equation. For the equation 2x plus 3y equals 12, enter 2 in the x field, 3 in the y field, and 12 in the constant field. Use negative numbers for subtraction.