Solve systems of linear equations using elimination, substitution, and matrix methods. Supports 2-variable and 3-variable systems with step-by-step solutions.
🔹 Elimination Method: Add or subtract equations to eliminate one variable, then solve for the remaining variables.
Problem: 3 notebooks and 2 pens cost $7.00. 5 notebooks and 1 pen cost $10.50. How much does each cost?
Equations: 3x + 2y = 7, 5x + y = 10.50
Multiply Eq 2 by 2: 10x + 2y = 21. Subtract Eq 1: 7x = 14 → x = $2.00 per notebook
Substitute: 5(2) + y = 10.50 → y = $0.50 per pen
Problem: A boat travels 60 km downstream in 3 hours and returns upstream in 5 hours. Find boat speed (x) and current speed (y).
System: x + y = 20, x - y = 12
Add equations: 2x = 32 → x = 16 km/h (boat speed)
Substitute: 16 + y = 20 → y = 4 km/h (current speed)
Problem: Mix 10% and 30% acid solutions to make 200 mL of 18% acid. How much of each?
Let x = mL of 10%, y = mL of 30%. System: x + y = 200, 0.10x + 0.30y = 36
Substitute y = 200 - x: 0.10x + 0.30(200 - x) = 36 → -0.20x = -24 → x = 120 mL of 10%, y = 80 mL of 30%
Problem: $12,000 invested at 4%, 5%, and 6%. Total interest is $620. The 5% amount is twice the 4% amount. Find each amount.
Let x = 4% amount, y = 5% amount, z = 6% amount.
System: x + y + z = 12000, 0.04x + 0.05y + 0.06z = 620, y = 2x
Substitute y = 2x: 3x + z = 12000, 0.14x + 0.06z = 620. Solve: x = $2,500 at 4%, y = $5,000 at 5%, z = $4,500 at 6%
A system of linear equations is a collection of two or more linear equations sharing the same variables. The solution is the set of values that satisfy all equations simultaneously.
If D = 0, the system has no unique solution. Use elimination to check if it's dependent or inconsistent.
Use elimination when coefficients are easy to match. Use substitution when a variable has coefficient 1 or -1.
Always plug the solution back into all original equations to catch arithmetic errors.
A 3-variable system represents three planes in 3D space. The solution is their intersection point.
A system of linear equations (simultaneous equations) is a set of two or more linear equations sharing the same variables. Solving it means finding values that satisfy every equation simultaneously. A 2-variable system represents two lines; their intersection is the solution. A 3-variable system represents three planes in 3D space.
Three primary methods are available. Choose the one best suited to your problem.
Multiply equations to align coefficients, then add/subtract to eliminate a variable. Solve the resulting equation, then back-substitute.
Example: 2x + 3y = 12, 4x - y = 10. Multiply Eq 2 by 3: 12x - 3y = 30. Add to Eq 1: 14x = 42 → x = 3. Then 2(3) + 3y = 12 → y = 2. Solution: (3, 2).
Solve one equation for one variable, then substitute into the other. Especially useful when a variable has coefficient 1 or -1.
Example: x + y = 10, 3x - 2y = 5. From Eq 1: y = 10 - x. Substitute: 3x - 2(10 - x) = 5 → 5x = 25 → x = 5, y = 5.
Calculate the determinant D of the coefficient matrix. Replace each variable column with constants to get Dx, Dy, Dz. Then x = Dx/D, y = Dy/D, z = Dz/D (when D ≠ 0).
Systems of equations are used across many fields:
Find break-even points, equilibrium price and quantity, and allocate resources across multiple constraints.
Solve circuit problems (Kirchhoff's laws), analyze forces in structures, and model chemical reactions.
Calculate speeds, distances, and times for multiple vehicles. Solve rate-of-work problems with multiple workers.
Mix solutions to target concentrations, calculate medication dosages, and balance chemical equations.
⚠️ Important Note: This System of Equations Solver is for educational and informational purposes only. While every effort has been made to ensure accuracy, results should be verified independently for critical applications such as engineering, scientific research, financial decisions, or medical calculations. Always consult a qualified professional for high-stakes decision-making.