How do I solve linear equations in two variables using Cramer's rule (determinant method)?

Question

Maharashtra Board SSC Algebra, Linear Equations in Two Variables: when I solve by Cramer's rule, how do I set up the determinants without mixing up the order?

GradeHunt_Team · Answer

Write both equations in the form a1x + b1y = c1 and a2x + b2y = c2. This step is where most mistakes happen, so do it first.

Then find three determinants:
D = | a1 b1 | = a1b2 - a2b1
    | a2 b2 |
Dx = | c1 b1 | = c1b2 - c2b1  (replace the x column with constants)
     | c2 b2 |
Dy = | a1 c1 | = a1c2 - a2c1  (replace the y column with constants)
     | a2 c2 |

Then x = Dx / D and y = Dy / D. This works only if D is not 0; if D = 0, the equations do not have a unique solution, so Cramer's rule cannot be used.

Example:
3x - 4y = 10
4x + 3y = 5
D = (3)(3) - (4)(-4) = 9 + 16 = 25
Dx = (10)(3) - (5)(-4) = 30 + 20 = 50
Dy = (3)(5) - (4)(10) = 15 - 40 = -25
x = 50/25 = 2, y = -25/25 = -1

Check: 3(2) - 4(-1) = 10 and 4(2) + 3(-1) = 5, so both equations are satisfied.

Common mistakes:
1. Not rearranging into ax + by = c first. If an equation is given as 2x = 5 - 3y, rewrite it as 2x + 3y = 5.
2. Sign errors in cross multiplication: always subtract (bottom-left times top-right) from (top-left times bottom-right).
3. Mixing up Dx and Dy. Remember: the constants replace the column of the variable you are finding.

Exam tip: show D, Dx and Dy separately with their values. Each value can then be checked on its own, and a slip at the last step does not hide the correct working before it. Always verify by substituting back into both equations if you have time.

How do I solve linear equations in two variables using Cramer's rule (determinant method)?

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