Solve 2x2 linear systems with step-by-step Cramer's rule working
Simultaneous equations are a core topic in GCSE maths, appearing across AQA, Edexcel, OCR, and WJEC exam boards. They involve finding values of two unknowns (usually x and y) that satisfy two equations at the same time. This skill is essential for the Algebra strand of the UK National Curriculum and forms the basis for more advanced work in A-level maths, physics, and engineering. Our simultaneous equations calculator uses Cramer's rule to solve 2x2 linear systems instantly. Simply enter the coefficients for both equations in the form ax + by = c, and the calculator returns the values of x and y along with the solution type. It handles all three possible outcomes: a unique solution (the lines cross at one point), no solution (the lines are parallel and never meet), and infinite solutions (the two equations describe the same line). Whether you are revising for GCSE maths exams, checking homework answers, or exploring how changing coefficients affects the solution, this tool provides instant results with clear labelling. Understanding simultaneous equations is also valuable in real-world contexts such as balancing budgets, mixing ingredients in specific ratios, and solving physics problems involving two unknowns.
To solve a pair of simultaneous equations: 1. Write your equations in the standard form: a1*x + b1*y = c1 (equation 1) and a2*x + b2*y = c2 (equation 2). For example, if you have 2x + 3y = 8 and x - y = 1, the coefficients are a1 = 2, b1 = 3, c1 = 8, a2 = 1, b2 = -1, c2 = 1. 2. Enter the coefficients for equation 1: set a (equation 1), b (equation 1), and the right-hand side value = (equation 1). 3. Enter the coefficients for equation 2: set a (equation 2), b (equation 2), and the right-hand side value = (equation 2). 4. Review the results. The calculator displays the values of x and y (rounded to 4 decimal places), the solution type, and the formatted equations. If the determinant is zero, the calculator identifies whether the system has no solution (parallel lines) or infinite solutions (same line). 5. Use the bar chart to visualise the x and y values when a unique solution exists.
This calculator solves 2x2 linear systems using Cramer's rule, a direct algebraic method based on determinants. Given two equations: a1*x + b1*y = c1 a2*x + b2*y = c2 Step 1: Calculate the determinant (D) of the coefficient matrix: D = a1*b2 - a2*b1 Step 2: If D is not zero, the system has a unique solution: x = (c1*b2 - c2*b1) / D y = (a1*c2 - a2*c1) / D Step 3: If D equals zero, check whether the equations are multiples of each other: If a1*c2 = a2*c1 AND b1*c2 = b2*c1, then the lines are identical (infinite solutions). Otherwise, the lines are parallel (no solution). The results are rounded to 4 decimal places to handle cases where the solution involves recurring decimals. This method is equivalent to solving by elimination or substitution but provides a direct formula that avoids intermediate steps. Cramer's rule is named after the Swiss mathematician Gabriel Cramer and works for any size system of linear equations, though it is most practical for 2x2 and 3x3 systems. For GCSE maths, the key takeaway is that a system of two linear equations either has exactly one solution, no solution, or infinitely many solutions.
Inputs: a1 = 2, b1 = 3, c1 = 8, a2 = 1, b2 = -1, c2 = 1
Inputs: a1 = 2, b1 = 4, c1 = 10, a2 = 1, b2 = 2, c2 = 3
Inputs: a1 = 3, b1 = 2, c1 = 12, a2 = 1, b2 = -1, c2 = 1