Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
A little investigation: at the bakery, two pens and one notebook cost 7 euros, and one pen with the same notebook costs 5 euros. How much does each item cost? One clue alone is not enough, but the two together reveal everything. Two unknowns require two equations: that is a system.
A pair (x, y) is a solution of a system when it satisfies BOTH equations at the same time. This is the classic quiz trap: a pair that works in only one equation is not a solution. Always check both!
First method, substitution: express one unknown in terms of the other using one equation, then replace it in the second. You are left with one equation and one unknown, and you know how to handle that.
Second method, elimination: add or subtract the equations to make one unknown disappear. For example, if one contains +y and the other -y, their sum eliminates y at once.
Geometrically, each equation draws a line, and solving the system means finding their intersection point. Two crossing lines: one unique solution. Two distinct parallel lines: no solution. Two identical lines: infinitely many solutions.
Solve by elimination: { x + y = 5, 2x - y = 1 }.
(x + y) + (2x - y) = 5 + 1
I add the two equations: the +y and the -y will cancel out.
3x = 6 donc x = 2
Only x remains, which I isolate by dividing by 3.
2 + y = 5 donc y = 3
I replace x with 2 in the first equation to find y.
2 × 2 - 3 = 1
Check in the second equation: 4 - 3 = 1, all good.
Result: The solution is the pair (2, 3).
Solve by substitution: { y = x + 1, x + 2y = 8 }.
x + 2(x + 1) = 8
The first equation gives me y: I substitute it into the second.
3x + 2 = 8 donc 3x = 6 et x = 2
I expand, simplify, and solve this single unknown equation.
y = 2 + 1 = 3
I go back to y using the first equation.
2 + 2 × 3 = 8
Check: 2 + 6 = 8, the pair (2, 3) satisfies both equations.
Result: The solution is the pair (2, 3).