How to use this solver
Enter coefficients in a consistent variable order, followed by each equation’s constant. Use zero for a missing variable. Gaussian elimination swaps, scales and combines rows until the solution can be read from the reduced matrix. All row operations here use exact fractions.
Questions about simultaneous equations
How are infinitely many solutions shown?
The calculator identifies free variables and expresses the remaining variables in terms of them. This describes a family of solutions.
Why is there sometimes no graph?
Graphs are shown for two-variable systems when both equations can be written as y = mx + b. Vertical lines and three-variable systems are explained using the matrix and solution checks.
Can I type a whole equation?
Use the labelled coefficient fields. First collect like terms and expand brackets yourself. This solver does not parse unrestricted equation text.
How can I calculate how many adult and child tickets were sold?
Suppose 20 tickets brought in $160, with adult tickets at $10 and child tickets at $5. Let x be adult tickets and y be child tickets. Enter the rows 1, 1, 20 and 10, 5, 160, representing x + y = 20 and 10x + 5y = 160. The solution is x = 12 and y = 8. Check: 12 + 8 = 20 and 12 × $10 + 8 × $5 = $160.
Can I find individual prices from the totals for two bundles?
Yes, provided the unit prices are the same in both purchases. Suppose two notebooks and three pens cost $16, while four notebooks and one pen cost $12. Let x be a notebook’s price and y a pen’s price. Enter 2, 3, 16 and 4, 1, 12. The solution is $2 per notebook and $4 per pen. Fixed fees, discounts or taxes must be accounted for consistently before setting up the equations.
How do I calculate the amounts needed for a mixture?
For a mathematical mixture example, suppose you want 10 L at 32% concentration using 20% and 50% solutions. Let x and y be their volumes in litres. Enter 1, 1, 10 and 0.2, 0.5, 3.2. The equations are x + y = 10 and 0.2x + 0.5y = 3.2. The solution is 6 L of the 20% solution and 4 L of the 50% solution. This model assumes compatible concentration units and that the volumes add directly.
What is a practical example with three unknowns?
Suppose a shipment contains 30 small, medium and large packages weighing 2 kg, 3 kg and 5 kg each. Its total weight is 98 kg, and there are two more small packages than large ones. Let x, y and z be their counts. Select three unknowns and enter rows 1, 1, 1, 30; 2, 3, 5, 98; and 1, 0, −1, 2. The solution is 12 small, 8 medium and 10 large packages.
How many facts do I need to find all the unknowns?
You normally need two independent linear equations for two unknowns, or three for three unknowns. Repeating the same relationship in a different form adds no new information. For example, x + y = 10 and 2x + 2y = 20 describe the same constraint, so they cannot determine x and y separately.
What does no solution mean when I use real measurements?
The entered relationships cannot all be true at once. Check for transcription errors, inconsistent units or missing charges. Rounded or noisy measurements can also produce contradictory equations. This calculator solves the entries exactly; it does not find a best-fit estimate for inconsistent data.
What if the solution gives a negative or fractional number of items?
A mathematical solution does not automatically meet practical restrictions. Ticket and package counts must normally be nonnegative whole numbers, while prices and measured quantities may be fractional. Do not round each answer and assume the totals still work. Recheck every original relationship and investigate the assumptions if the counts are impossible.
Can this solve problems where one unknown is multiplied by another?
No. Each equation must be linear in the unknowns. Terms such as 3x and −2y are allowed, but xy, x² and 1/x are not. For a rectangular area problem, length × width is nonlinear unless one dimension is already known. Keep the same variable order in every row and enter zero for a missing variable.