How To Solve A System Of Linear Equations Calculator

The formula this calculator uses

D = a1 b2 - a2 b1

a1, b1, c1

The coefficients and constant of the first equation

a2, b2, c2

The coefficients and constant of the second equation

D

The determinant of the coefficient matrix

x, y

The values that satisfy both equations at once

How to check the result by hand

1. 1

   Write both equations in standard form

   Each equation must read ax + by = c, with the unknowns on the left in that order and the constant alone on the right. Rearranging first prevents coefficients from being read off in the wrong order.

2. 2

   Build the determinant from the coefficients

   Multiply the first x coefficient by the second y coefficient and subtract the second x coefficient times the first y coefficient. For 2x + 3y = 13 and 4x - y = 5 that is 2 times -1 minus 4 times 3, giving -14.

3. 3

   Check that the determinant is not zero

   A zero determinant means the lines are parallel or identical, so there is no single solution. Comparing the ratio of the x coefficients with the ratio of the y coefficients reveals the parallel case before any division is attempted.

4. 4

   Replace the x column with the constants

   Keep the y coefficients where they are and put the two constants in the x column. The numerator is c1 b2 minus c2 b1, which for the example is -28, and dividing by -14 gives x = 2.

5. 5

   Replace the y column and check the answer

   Swap the constants into the y column for a numerator of a1 c2 minus a2 c1, giving y = 3. Substitute both values back into the original equations; each must return its stated constant.