Simultaneous equations

Simultaneous equations are two or more equations solved together to find values that satisfy all of them at once. The substitution method rearranges one equation to isolate a variable, then substitutes that into the other equation. The elimination method adds or subtracts the equations to cancel out one variable, making the other easy to solve for.

Key terms

Simultaneous equations:

Two or more equations solved together for shared solutions.

Substitution:

Solving by replacing a variable with an equivalent expression.

Elimination:

Solving by adding or subtracting equations to cancel a variable.

Questions

1. 1. Simultaneous equations are:

   • Two or more equations solved together
   • A single equation with no unknowns
   • Always unsolvable
   • The same as one equation repeated

2. 2. The elimination method works by:

   • Adding or subtracting equations to cancel a variable
   • Ignoring one of the equations completely
   • Always multiplying by zero
   • Removing all variables at once

3. 3. The substitution method works by:

   • Replacing a variable with an equivalent expression
   • Adding two equations together only
   • Ignoring all variables
   • Guessing randomly

4. 4. Solve: x + y = 10, x − y = 2 (find x)

   • 6
   • 8
   • 10
   • 2

5. 5. Using the same equations (x + y = 10, x − y = 2), what is y?

   • 4
   • 6
   • 8
   • 10

6. 6. Solving simultaneous equations finds values that:

   • Satisfy all the equations at once
   • Satisfy only one of the equations
   • Satisfy no equations
   • Are always zero

7. 7. Two straight-line equations solved simultaneously find their:

   • Point of intersection
   • Individual y-intercepts only
   • Individual gradients only
   • Nothing meaningful

8. 8. Solve: 2x + y = 12, x = 3 (find y using substitution)

   • 6
   • 9
   • 12
   • 3

9. 9. Solve: x + 2y = 8, x − y = 2 (find x)

   • 4
   • 2
   • 6
   • 8

10. 10. Using the same equations (x + 2y = 8, x − y = 2), what is y?

    • 2
    • 4
    • 6
    • 1

11. 11. Solve: 3x + y = 15, y = 2x (find x)

    • 3
    • 5
    • 15
    • 2

12. 12. Two mobile phone plans: Plan A costs 20 + 0.5g, Plan B costs 10 + 0.75g (g = gigabytes). Setting them equal finds:

    • The number of gigabytes where both plans cost the same
    • The total combined cost of both plans
    • Nothing useful
    • The cost with zero gigabytes

13. 13. Solve: 2x − y = 4, x + y = 5 (find x)

    • 3
    • 2
    • 5
    • 4

14. 14. Why is the elimination method useful when neither equation has an isolated variable?

    • It avoids the extra step of rearranging an equation before solving
    • Elimination never works unless a variable is already isolated
    • Elimination and substitution always require identical steps
    • Elimination cannot be used unless both equations are identical

15. 15. You buy 3 apples and 2 bananas for $7, and 1 apple and 4 bananas for $9. Setting up simultaneous equations would help find:

    • The individual price of an apple and a banana
    • Only the total number of fruits bought
    • Nothing useful about the prices
    • The total combined weight

16. 16. Solve: 4x + 3y = 18, 2x + y = 8 (find x and y)

    • x = 3, y = 2
    • x = 2, y = 4
    • x = 4, y = 0
    • x = 1, y = 6

17. 17. Two candles burn at different rates: Candle A starts at 20cm and burns 2cm/hour; Candle B starts at 15cm and burns 1cm/hour. When are they equal height?

    • After 5 hours
    • After 10 hours
    • After 2 hours
    • They are never equal

18. 18. Solve: x + y = 7, 2x − 3y = -1 (find x and y)

    • x = 4, y = 3
    • x = 3, y = 4
    • x = 7, y = 0
    • x = 1, y = 6

19. 19. A system of two simultaneous linear equations with no solution represents two lines that:

    • Are parallel and never intersect
    • Always intersect at exactly one point
    • Are identical, overlapping lines
    • Are perpendicular

20. 20. A system of two simultaneous linear equations with infinite solutions represents two lines that:

    • Are actually the same line
    • Are parallel and distinct
    • Intersect at exactly one point
    • Never touch at all

21. 21. Solve: 5x − 2y = 16, 3x + 2y = 16 (find x and y)

    • x = 4, y = 2
    • x = 2, y = 4
    • x = 8, y = 0
    • x = 0, y = 8