Simultaneous Equations – Two Unknowns, One Solution
Simultaneous equations are a pair of equations containing two unknown variables. The solution is a single pair of values that satisfies both equations at the same time.
Why Two Equations?
One equation with two unknowns has infinitely many solutions. A second equation with the same two unknowns narrows the possibilities to exactly one point — the pair of values that works for both simultaneously.
Method 1 – Elimination
Add or subtract the equations to eliminate one variable, solve for the other, then substitute back.
Add the equations: 2x = 10, so x = 5. Substitute: 5 + y = 7, so y = 2. Answer: x = 5, y = 2. Check in both: 5+2=7 and 5-2=3. Correct.
Subtract equation 2 from equation 1: 2x = 8, so x = 4. Substitute: 4 + 2y = 8, so 2y = 4, y = 2. Answer: x = 4, y = 2.
Multiply equation 2 by 3: 12x - 3y = 15. Add to equation 1: 14x = 28, so x = 2. Substitute into eq 2: 8 - y = 5, so y = 3. Answer: x = 2, y = 3.
Method 2 – Substitution
Rearrange one equation to make one variable the subject, then substitute into the other equation.
Substitute y = 2x - 1 into 3x + y = 9: 3x + (2x - 1) = 9. 5x - 1 = 9. 5x = 10. x = 2. Then y = 2(2) - 1 = 3. Answer: x = 2, y = 3.
Real-Life Application
Two friends buy coffees and teas. Anna buys 2 coffees and 1 tea for 7. Ben buys 1 coffee and 2 teas for 5. Find the price of each.
Let c = coffee price, t = tea price. 2c + t = 7 and c + 2t = 5. From equation 2: c = 5 - 2t. Substitute: 2(5-2t) + t = 7. 10 - 4t + t = 7. -3t = -3. t = 1. c = 5 - 2 = 3. Coffee = 3, Tea = 1.
Common Mistakes
- Forgetting to check the solution in both original equations.
- Making sign errors when subtracting equations.
- Multiplying only part of an equation when scaling to match coefficients.
Key Takeaways
- Simultaneous equations have two unknowns and two equations — together they give one unique solution.
- Elimination: scale the equations so one variable cancels when you add or subtract.
- Substitution: express one variable in terms of the other and substitute.
- Always check your solution in both original equations.
Practice Questions
- Solve: x + y = 10 and x - y = 4.
- Solve: 2x + y = 11 and x + 3y = 13.
- Solve by substitution: y = x + 2 and 3x + 2y = 14.
- Tickets for a show cost 8 for adults and 5 for children. A group of 10 people paid 65 total. How many adults and children were there?
- Solve: 3a - 2b = 4 and 5a + b = 21.