Solving Systems of Two Equations: Substitution, Elimination, and Word Problems
October 29, 2026
A system of equations is a set of equations that share variables, and the solution is the pair of values that satisfies all of them. Real-life problems with two unknown quantities and two pieces of information become systems of two equations.
Example: a theater sells adult tickets at $12 and child tickets at $7. It sold 100 tickets for $1,050 in total. Let a be the number of adult tickets and c the number of child tickets. Then a + c = 100 and 12a + 7c = 1,050.
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Open the Scientific Calculator →Substitution
Solve one equation for one variable, then substitute into the other. From a + c = 100, c = 100 − a. Substituting into the second: 12a + 7(100 − a) = 1,050, so 12a + 700 − 7a = 1,050, 5a = 350, a = 70. Then c = 30. Check: 70 × 12 = 840, 30 × 7 = 210, total 1,050. ✓. Substitution works well when one equation already has a variable by itself or with a coefficient of 1.
Elimination
Add or subtract the equations (after scaling) so one variable cancels. Multiply the first equation by 7: 7a + 7c = 700. Subtract it from the second: (12a + 7c) − (7a + 7c) = 1,050 − 700, giving 5a = 350, a = 70. Elimination is useful when both equations are in standard form. For a mixture problem: how many liters of a 30% solution and a 50% solution make 10 liters of a 40% solution? x + y = 10 and 0.30x + 0.50y = 4. Multiply the first by 0.30: 0.30x + 0.30y = 3; subtract: 0.20y = 1, y = 5, x = 5.
| Problem | Equations | Solution |
|---|---|---|
| Tickets: $12 and $7, 100 tickets, $1,050 | a + c = 100; 12a + 7c = 1,050 | a = 70, c = 30 |
| Mixture: 30% and 50% to make 10 L of 40% | x + y = 10; 0.3x + 0.5y = 4 | x = 5, y = 5 |
| Phone plans: $20 + $0.10/min vs. $0.15/min | y = 20 + 0.10x; y = 0.15x | x = 400 minutes, y = $60 |
Break-Even With Two Plans
Plan A costs $20 a month plus $0.10 per minute. Plan B has no monthly fee and charges $0.15 per minute. Set the costs equal: 20 + 0.10x = 0.15x, so 0.05x = 20, and x = 400 minutes. At 400 minutes, both cost $60. Below 400 minutes, plan B is cheaper; above it, plan A. Graphically, the solution is the intersection of the two lines. This is the same logic used for break-even analysis in business.
Special Cases and Checking
If the lines are parallel (same slope, different intercepts), there is no solution; if the equations describe the same line, there are infinitely many solutions. In elimination, this appears as 0 = a nonzero number (no solution) or 0 = 0 (infinitely many). Always substitute your answer back into both original equations. Make sure that the answers make sense in the context: counts of tickets must be whole, non-negative numbers.
Frequently Asked Questions
What is a system of equations?
Two or more equations that share unknowns and are solved together.
When should I use substitution?
When one variable is already isolated or easy to isolate.
When should I use elimination?
When the equations are in standard form and coefficients can be matched.
What does it mean if there is no solution?
The lines are parallel and never intersect.
How do I check my answer?
Substitute the values into both original equations.