AI System of Equations Solver
A system of equations has two or more equations that must be true at once. This AI system of equations solver chooses between substitution and elimination, shows each combination and checks the answer in both equations.
Substitution versus elimination
Substitution works best when one equation already has a letter on its own, such as y = 2x − 1. Replace that letter in the other equation and solve what remains. Elimination works best when the equations are in the same layout: add or subtract them so one letter cancels. Multiply an equation first if the coefficients do not match.
What the solution means graphically
Each linear equation is a straight line. The solution to the system is the point where the lines cross. If the lines are parallel, there is no solution; if they are the same line, there are infinitely many. Visualise this with the graphing solver.
Where systems appear
Systems appear in mixture problems, ticket and price questions and break-even analysis. If you are building the equations from a story, begin with the word problem solver, and for a single equation use the linear equation solver.
Worked examples
Example 1: Solve x + y = 10 and x − y = 2 (elimination)
- Add the equations: 2x = 12, so x = 6.
- Substitute into x + y = 10: y = 4.
- Check: 6 − 4 = 2 ✓
Final answer: x = 6, y = 4
Example 2: Solve y = 2x − 1 and 3x + y = 14 (substitution)
- Substitute y: 3x + 2x − 1 = 14, so 5x = 15 and x = 3.
- Then y = 2(3) − 1 = 5.
Final answer: x = 3, y = 5
Example 3: Solve 2x + 3y = 13 and 4x − y = 5
- Multiply the second equation by 3: 12x − 3y = 15.
- Add to the first: 14x = 28, so x = 2.
- Substitute: 4 + 3y = 13, so y = 3.
- Check: 4(2) − 3 = 5 ✓
Final answer: x = 2, y = 3
Example 4: Solve x + y = 2 and x + y = 5
- Subtract the first equation from the second: 0 = 3.
- This is impossible, so the lines are parallel.
Final answer: No solution
Common mistakes to avoid
- Subtracting equations without changing every sign of the second one.
- Solving for only one variable and stopping.
- Substituting back into the wrong equation after rearranging.
- Forgetting to multiply every term of an equation when scaling it.
- Not checking the pair in both equations.
Frequently asked questions
Which method should I choose?
Use substitution when one variable is already isolated; otherwise elimination is usually faster.
Can it solve three equations with three unknowns?
Yes. Type them in AI mode and it will reduce the system step by step.
What does “inconsistent” mean?
A system with no solution, because the equations contradict each other.
Can the solution be a fraction?
Yes. Exact fractions are kept, such as x = 7/3, rather than rounding too early.
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AI-generated solutions are for learning purposes. Always verify important answers.