Substitution
Make the easiest unknown the subject (one with a coefficient of 1 when there is one) and substitute it into the other equation, with its brackets and, where needed, its denominator.
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Teaching Guide: Solving Systems of Equations
Systems of two equations in two unknowns, or simultaneous equations, are the first step into algebra with several unknowns: students solve them by substitution and elimination, then by drawing the two lines, and learn to tell when a system has no solution or infinitely many. This free systems of equations worksheet generator builds A4 worksheets of systems made from their solution, with an answer key worked step by step in the method you choose.
Student worksheet 
Answer key Equalisation
Make the same unknown the subject of both equations, set the two expressions equal and solve the linear equation that results. It is how Spanish schools teach the comparison method.
Elimination
Multiply the equations so that one unknown has opposite coefficients, then add them and it disappears. The answer key sets out the sum in columns.
Graphical method
Tables of values with x values that give whole-number y values, both lines on axes from −6 to 6 and the crossing point. At level 3, parallel lines and the same line too.
Classify
One solution, infinitely many or none from a/a′, b/b′ and c/c′, also after simplifying the equations, and the value of the parameter k.
Word problems
Heads and legs, banknotes, tickets, perimeters, ages, digits and mixtures: the answer key names the unknowns, sets up the system, solves it and answers the question.
Frequently Asked Questions about Systems of Equations
Do the solutions come out exactly?
Yes. Each system is built from its solution (whole numbers or, at level 3, fractions) and the generator solves it again step by step with exact arithmetic in the chosen method. It is only used if it reaches exactly that solution.
Can I choose the method used in the answer key?
Yes: the first step of the generator is the method (substitution, equalisation or elimination) and the whole answer key follows it. You can make a worksheet on one method or mix all three on one sheet, one per exercise.
Which school years are they for?
The three methods with simple coefficients and direct word problems suit Grade 8; any coefficients, the graphical method and classifying systems, Grade 9; equations to simplify first, the parameter k and age and mixture problems, Grade 10. Students should be confident with linear equations first.
Can I mix systems with other exercises?
Yes. Add one exercise for each method and level, or from other generators such as quadratic equations or proportionality, and they all share the worksheet and the answer key. Save it under Templates to use it again.
Is any student data sent to a server?
No. Every system is generated in your browser and no student data is stored.
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