Simultaneous Equations

Q: Solve x + y = 7 and x − y = 3.

x=5, y=2 — Add the equations: 2x = 10, so x = 5. Substitute back: 5 + y = 7, so y = 2.

Q: Solve 2x + y = 8 and x − y = 1.

x=3, y=2 — Add the equations: 3x = 9, so x = 3. Substitute back: 3 − y = 1, so y = 2.

Q: An adult ticket costs ¥x and a child ticket ¥y. Two adults and one child pay ¥2,500; one adult and two children pay ¥2,000. What does an adult ticket cost?

¥1,000 — The equations are 2x + y = 2500 and x + 2y = 2000. Eliminating y gives 3x = 3000, so x = 1000.

Q: Solve y = 2x and x + y = 9.

x=3, y=6 — Substitute y = 2x into the second equation: x + 2x = 9, so 3x = 9 and x = 3. Then y = 6.

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

x=4, y=2 — Both equations have 2y, so subtract: 2x = 8 and x = 4. Substitute back: 4 + 2y = 8, so y = 2.

Q: Solve x + 2y = 10 and 2x + y = 8.

x=2, y=4 — Double the first equation and subtract the second: 3y = 12, so y = 4, and then x = 2.

Q: You have 12 coins in total, some ¥100 and some ¥50, worth ¥950 altogether. How many ¥100 coins are there?

7 coins — Let x be the number of ¥100 coins and y the ¥50 coins: x + y = 12 and 100x + 50y = 950. Subtracting 50 times the first equation leaves 50x = 350, so x = 7.

Q: Solve 4x − y = 5 and 2x + y = 7.

x=2, y=3 — The y terms cancel when you add: 6x = 12, so x = 2. Substitute into 2x + y = 7: 4 + y = 7, so y = 3.

Where this is used

Two adults and three children pay ¥3,600 for tickets. Three adults and two children pay ¥3,800. What does one of each ticket cost? Simultaneous equations are the tool for pinning down values that satisfy several conditions at the same time. Pricing, factory production plans, even traffic flow through a network — whenever two constraints have to hold together, this is how you solve them.

Start gently

The x and y that satisfy two equations at once are the point where the two lines cross. Solving the equations and finding the intersection are the same act.

Key points

What Simultaneous Equations Are

The task is to find the values of x and y that make two or more equations true at the same time. One equation on its own has endless solutions; add a second and, in general, the answer narrows down to a single pair.

Elimination (Add or Subtract to Cancel)

Match the coefficients of one letter, then add or subtract the equations so that letter disappears. If the coefficients already have the same size, you can add or subtract straight away.

Substitution (Swap One Letter In)

Rearrange one equation into the form "y = something", then put that something into the other equation. If one equation is already written as "y = …", you can substitute immediately.

Try It on the Ticket Prices

Let an adult ticket be ¥x and a child ticket ¥y. Two adults plus three children come to 3600, three adults plus two children to 3800. Solve the pair and you get ¥1000 for an adult and ¥800 for a child.

See it drawn

The solution is where x + y = 5 meets 2x − y = 1

The intersection (2, 3) is the solution of the system.

Check yourself 10questions

Tap an option to check

Q1 Solve x + y = 7 and x − y = 3.

Q2 Solve 2x + y = 8 and x − y = 1.

Q3 An adult ticket costs ¥x and a child ticket ¥y. Two adults and one child pay ¥2,500; one adult and two children pay ¥2,000. What does an adult ticket cost?

Q4 Solve y = 2x and x + y = 9.

Q5 Solve 3x + 2y = 16 and x + 2y = 8.

Q6 Solve x + 2y = 10 and 2x + y = 8.

Q7 You have 12 coins in total, some ¥100 and some ¥50, worth ¥950 altogether. How many ¥100 coins are there?

Q8 Solve 4x − y = 5 and 2x + y = 7.

Q9 Three pencils and two notebooks cost ¥460; two pencils and three notebooks cost ¥540. What does one notebook cost?

Q10 When does a pair of simultaneous equations have infinitely many solutions?

Jobs that use this

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