Level 3 · Ages 11–13
Imagine you are buying cinema tickets at 3 euros each. For 2 tickets you pay 3 × 2 = 6 euros. For 5 tickets, 3 × 5 = 15 euros. And for a number of tickets you do not know yet? Mathematicians have a brilliant trick: they replace the unknown number with a letter, for example n. The price becomes 3 × n, which we simply write 3n.
An expression like 3n or 2x + 5 is called a literal expression: it contains letters that stand for numbers. The letter can stand for a number you do not know yet, or a number that changes with the situation. Two writing conventions to know: we drop the × sign before a letter (3 × n is written 3n) and x × x is written x².
Substituting means replacing the letter with a specific value, then computing. For example, if x = 4, then 2x + 5 equals 2 × 4 + 5 = 13. Careful: you put the × sign back when you compute, and you follow the order of operations (multiplication before addition).
Simplifying an expression means making it shorter by grouping what belongs together. 4x + 3x is 4 packs of x plus 3 packs of x, so 7 packs: 7x. However, 5x + 2 cannot be simplified: 5x and 2 are not the same kind, like apples and oranges. Your turn!
Read it again any time, including while you work through the exercises.
Imagine you are buying cinema tickets at 3 euros each. For 2 tickets you pay 3 × 2 = 6 euros. For 5 tickets, 3 × 5 = 15 euros. And for a number of tickets you do not know yet? Mathematicians have a brilliant trick: they replace the unknown number with a letter, for example n. The price becomes 3 × n, which we simply write 3n.
An expression like 3n or 2x + 5 is called a literal expression: it contains letters that stand for numbers. The letter can stand for a number you do not know yet, or a number that changes with the situation. Two writing conventions to know: we drop the × sign before a letter (3 × n is written 3n) and x × x is written x².
Substituting means replacing the letter with a specific value, then computing. For example, if x = 4, then 2x + 5 equals 2 × 4 + 5 = 13. Careful: you put the × sign back when you compute, and you follow the order of operations (multiplication before addition).
Simplifying an expression means making it shorter by grouping what belongs together. 4x + 3x is 4 packs of x plus 3 packs of x, so 7 packs: 7x. However, 5x + 2 cannot be simplified: 5x and 2 are not the same kind, like apples and oranges. Your turn!
Compute the value of 3x + 5 for x = 4.
3x + 5 = 3 × 4 + 5
I replace x with 4 and put back the hidden × sign.
3 × 4 = 12
Multiplication first: I compute 3 × 4.
12 + 5 = 17
All that is left is to add 5.
Result: For x = 4, the expression 3x + 5 equals 17.
Simplify the expression 4x + 7 + 3x + 2.
4x + 3x = 7x
I group the x terms: 4 packs of x plus 3 packs of x make 7 packs of x.
7 + 2 = 9
I group the plain numbers: 7 plus 2 make 9.
4x + 7 + 3x + 2 = 7x + 9
I put it all together. We cannot go further: 7x and 9 are not the same kind.
Result: 4x + 7 + 3x + 2 = 7x + 9