Pairs of linear equations are a core Class 9 and 10 topic and appear throughout physics and economics problems. Enter the coefficients of both equations to solve for x and y and see each determinant in Cramer's rule.
Write both equations as ax + by = c.
Enter the six coefficients.
Read x and y with the determinants.
D: D = a₁b₂ − a₂b₁
x: x = (c₁b₂ − c₂b₁) ÷ D
y: y = (a₁c₂ − a₂c₁) ÷ D
Two unknowns need two facts to pin them down. The simultaneous equations calculator solves a pair of linear equations of the form a₁x + b₁y = c₁ and a₂x + b₂y = c₂ using Cramer's rule. It shows the three determinants D, Dx and Dy, and then x = Dx ÷ D and y = Dy ÷ D.
Pair of linear equations in two variables is a full chapter in the Class 10 NCERT syllabus, where substitution, elimination and cross-multiplication are taught. Word problems about ages, the cost of two items, boat speeds upstream and downstream, or mixtures all reduce to such a pair. The calculator also tells you when there is no unique answer: when the two lines are parallel and never meet, or when both equations describe the same line.
1. Write both equations in the form ax + by = c, with the constants on the right.
2. Compute the coefficient determinant D = a₁b₂ − a₂b₁.
3. If D = 0, stop: the lines are parallel (no solution) or identical (infinitely many solutions).
4. Compute Dx = c₁b₂ − c₂b₁ by replacing the x-coefficients with the constants.
5. Compute Dy = a₁c₂ − a₂c₁ by replacing the y-coefficients with the constants.
6. Find x = Dx ÷ D and y = Dy ÷ D.
7. Substitute x and y back into both original equations to check.
Eliminate y by multiplying the first equation by b₂ and the second by b₁ and subtracting. The y terms cancel and you get (a₁b₂ − a₂b₁)x = c₁b₂ − c₂b₁, so x = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁). Eliminating x in the same way gives y = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁). The common denominator is the determinant D. Cramer's rule is simply elimination done once in general form, which is why it always agrees with other methods.
Each linear equation is a straight line in the xy-plane, and the solution is the point where they cross. D = 0 means the ratios a₁/a₂ and b₁/b₂ are equal, so the lines have the same slope. If c₁/c₂ matches too, they are the same line and every point on it works. If it does not, the lines are parallel and never meet. NCERT calls these consistent, dependent and inconsistent pairs, and the ratio test is the same idea as checking D.
When D is very close to zero, the lines are nearly parallel. A tiny change in one coefficient, perhaps from rounding a measured value, can move the crossing point a long way. Such systems are called ill-conditioned. In exam problems the numbers are chosen to avoid this, but with real data it is worth checking whether small input changes swing the answer wildly. The calculator treats only an exact zero as singular.
A stationery shop sold 36 pens in a day, some at ₹30 and the rest at ₹50, and collected ₹1,400. Tanvi writes x + y = 36 and 30x + 50y = 1400 to find how many of each were sold.
Equations: 1x + 1y = 36 30x + 50y = 1400
Determinant D: D = a₁b₂ − a₂b₁ = 1×50 − 30×1 = 20
Dx and Dy: Dx = c₁b₂ − c₂b₁ = 400 Dy = a₁c₂ − a₂c₁ = 320
Cramer's rule: x = Dx ÷ D = 400 ÷ 20 = 20 y = Dy ÷ D = 320 ÷ 20 = 16
Answer: Solution x = 20, y = 16; Determinant D 20
Leaving an equation as ax + by + c = 0 and entering c without flipping its sign.
Entering the coefficients of the second equation in the wrong order, such as swapping a₂ and b₂.
Swapping the order of subtraction in Dx or Dy, which flips the sign of the answer.
Concluding there is no solution whenever D = 0, without checking whether the lines are identical.
Using this method for equations with x², xy or 1/x terms, which are not linear.
Solving Class 10 word problems on ages, prices, speeds and fractions.
Finding the intersection point of two straight lines in coordinate geometry.
Mixture and alloy problems, such as blending two grades to hit a target.
Simple supply and demand models in economics.
Solving for two currents or voltages in small circuit problems.
What if D is zero?
The system has no unique solution: either no solution (parallel lines) or infinitely many (the same line).
Can I use substitution instead?
Yes. All methods give the same answer; Cramer's rule is simply the quickest to show step by step.