The worksheets themselves are in German – just as in class. The editor is only available in German so far.
Simultaneous equations – worksheets to print
Solve systems of two linear equations in two variables: by equating, by substitution and by elimination, read the intersection of two lines, tell how many solutions there are, and word problems such as ticket prices or hens and rabbits.
- Choose the number of tasks and the difficulty yourself
- With an answer sheet and, if you like, a QR code to the answers
- At three levels (★ to ★★★) and as group A and B
- As a single worksheet or as a task on a mixed worksheet
Simultaneous equations: explained simply
Two equations with the unknowns x and y belong together. You look for the pair of numbers that makes both equations true at once – on a graph, the point where two lines cross.
How to work it out
- Equating: if both equations read y = …, set the right-hand sides equal.
- Substitution: if one equation is solved for y (or x), put that expression into the other equation.
- Elimination: add or subtract the equations so that one variable drops out – multiply an equation by a number first if needed.
- Check: put x and y into both equations.
An example, step by step
Solve: I: x + y = 10, II: x − y = 2.
- Add I and II: 2x = 12, so x = 6.
- Put x = 6 into I: 6 + y = 10, so y = 4.
- Check: 6 + 4 = 10 and 6 − 4 = 2. Solution: x = 6, y = 4.
Understand other variations
Equating
y = 2x + 1 and y = −x + 7: 2x + 1 = −x + 7, so 3x = 6 and x = 2. Then y = 2 · 2 + 1 = 5.
Substitution
y = 3x and 2x + y = 10: 2x + 3x = 10, so 5x = 10, x = 2 and y = 6.
Multiply first
3x + 2y = 12 and x + 4y = 14: I · 2 gives 6x + 4y = 24. Subtract II: 5x = 10, so x = 2. From I: 6 + 2y = 12, so y = 3.
Number of solutions
x + y = 4 and 2x + 2y = 8 describe the same line: infinitely many solutions. x + y = 4 and x + y = 6 are parallel: no solution.
Word problems
20 heads and 56 legs: x + y = 20 and 2x + 4y = 56. With x = 20 − y you get 40 − 2y + 4y = 56, so y = 8 rabbits and x = 12 hens.
Your turn
Solve: I: y = x + 3, II: y = 2x + 1.
Show the answer and method
x + 3 = 2x + 1, so x = 2 and y = 5.
Ready-made worksheets: Simultaneous equations
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