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.

Systematic methods for solving simultaneous equations go back much further than modern algebra notation — the ancient Chinese mathematical text The Nine Chapters on the Mathematical Art (compiled over centuries, with a version dating to around 200 BCE) already describes a systematic elimination method for solving multiple linear equations together, remarkably similar to the elimination method taught today. Simultaneous equations model any situation with two unknowns and two independent constraints — from finding the price of two items given two different combinations bought, to working out a chemical mixture that must satisfy two conditions at once.

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.

Solve: x + y = 7 and x - y = 3.

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.

Solve: 3x + 2y = 16 and x + 2y = 8.

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.

Solve: 2x + 3y = 13 and 4x - y = 5.

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.

Solve: y = 2x - 1 and 3x + y = 9.

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.

Set up and solve.

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: Simultaneous Equations

Solve by Elimination