Calculator
Example data table
| Story detail | Equation value | Meaning |
|---|---|---|
| Total tickets sold | 1x + 1y = 31 | Adult tickets plus child tickets equal 31. |
| Total ticket revenue | 5x + 3y = 121 | Adult revenue plus child revenue equals 121. |
| Solution | x = 14, y = 17 | The theater sold 14 adult tickets and 17 child tickets. |
Formula used
The calculator solves two linear equations:
a1x + b1y = c1
a2x + b2y = c2
Main determinant:
D = a1b2 - a2b1
When D is not zero:
x = (c1b2 - c2b1) / D
y = (a1c2 - a2c1) / D
Residual checks use ax + by - c. A residual near zero confirms the solution.
How to use this calculator
- Read the word problem and name both unknown values.
- Enter labels for the first and second unknowns.
- Convert the first story fact into ax + by = c.
- Convert the second story fact into ax + by = c.
- Add an optional total expression for cost, weight, or value.
- Press submit to see the result above the form.
- Download the result as CSV or PDF for records.
Linear system word problem guide
A linear system word problem describes two unknown quantities. The story gives two relationships. Each relationship becomes one equation. The calculator solves both equations together. It also explains the determinant, substitution check, and final meaning.
Why these problems matter
Many classroom and business questions compare two items. Tickets, coins, mixtures, distance, prices, labor, and inventory all fit this model. A system helps you turn plain language into numbers. One equation may count total items. The other may count total value, weight, cost, or time.
Building the equations
Start by naming the unknowns. Let x represent the first item. Let y represent the second item. Next, find two facts from the story. Write each fact as a linear equation. The calculator uses the form ax + by = c. Enter the coefficients for both equations. A missing variable has a coefficient of zero. A negative coefficient is allowed.
Reading the answer
The determinant decides the solution type. A nonzero determinant gives one ordered pair. This means the story has one clear answer. A zero determinant needs extra checking. The system may have no solution. It may also have infinitely many solutions. In a word problem, no solution usually means the details conflict.
Advanced checks
The calculator also tests residuals. Residuals show how closely the answer satisfies each equation. Small residuals mean the algebra is consistent. Nonnegative checks are useful for real quantities. Tickets, people, coins, and products cannot be negative. When a result is negative, the equations may be valid algebraically, but the story may need review.
Using exports
Use the CSV file for spreadsheets or records. Use the PDF file for a clean study sheet. Both exports include equations, values, status, and steps. Keep the word problem text short and direct. Clear input creates clearer explanations.
Best input habits
Use consistent units in every equation. Do not mix dollars and cents unless you convert first. Round only after solving, not before. If a word problem gives more than two facts, choose the two facts that describe different relationships. Then use the extra fact as a check. This habit prevents copied coefficients and improves accuracy. Save your work after each attempt for later comparison and teacher review notes.
FAQs
What is a linear system word problem?
It is a story problem with two or more unknown values. This calculator focuses on two unknowns and two linear equations.
What does the determinant show?
The determinant shows whether the equations meet at one point. A nonzero determinant means one unique solution exists.
Can I use decimals in the equations?
Yes. You can enter decimals, negative numbers, or zero coefficients. Use consistent units for the best result.
What does no solution mean?
No solution means the two equations conflict. In a word problem, one fact may be copied incorrectly.
What do infinitely many solutions mean?
It means both equations describe the same relationship. The problem needs another independent fact to find exact values.
Why are residuals included?
Residuals check the answer by substituting it back into each equation. Values near zero show a strong match.
Can this solve mixture problems?
Yes. Enter one equation for total amount. Enter another equation for total concentration, value, or ingredient amount.
Can I export my work?
Yes. Use the CSV option for spreadsheet data. Use the PDF option for a simple printable solution sheet.