Equivalent means equal at every x — one substitution can refute, none can prove
Equal at every x
Two expressions are equivalent when they give the same value for every value of the letter. 3(x + 2) and 3x + 6 are equivalent: at x = 1 both are 9, at x = 10 both are 36, and there is no value of x where they differ.
Expanding, factoring and collecting like terms all turn an expression into an equivalent one. They change how it is written, never its value. That is why they are allowed.
A rectangle 3 tall and x + 2 long. It is made of a 3 by x piece and a 3 by 2 piece, so its area is 3x + 6.
The near-miss
3x + 2 looks close to 3(x + 2), but it is not equivalent. The 3 multiplies everything inside the bracket, so the 2 becomes 3 × 2 = 6. In 3x + 2 the 2 was never multiplied by 3.
One value is enough to show it. At x = 1, 3(x + 2) = 3 × 3 = 9, but 3x + 2 = 5. The two expressions disagree at x = 1, so they are not equivalent.
Squaring a sum
Another near-miss is . Write the square as two brackets and multiply out: . The middle term, 6x, is real.
At x = 1, and , but . So is not equivalent to .
A square of side x + 3 holds , 9, and two strips of 3x. Writing leaves out both strips.
One match proves nothing
Agreeing at one value is not enough. and 2x are both 4 when x = 2. But at x = 3, and 2x = 6, so they are not equivalent. Two different expressions can meet at a value or two and still be different everywhere else.
So substitution works in only one direction. One value where two expressions disagree proves they are not equivalent. Values where they agree prove nothing, however many you try. To show that two expressions are equivalent, use algebra: expand or simplify both until they are written the same way.
Find every x where the two sides agree
The curve is and the straight line is 2x. Slide x and read both values. They agree at x = 0 and at x = 2, and nowhere else.
Algebraic expressions · Three Pupils' Expressions for the Tiles Round a Square Pond
QuestionA square pond is n tiles long on each side, and one row of square tiles is laid all the way round it. Aisha says the border needs (n + 2)2 − n2 tiles. Ben says it needs 4(n + 1) tiles. (a) Show that the two expressions are equivalent. (b) Chen's expression is n2 + 8. It gives 12 tiles when n = 2, and so do the other two. Is Chen's expression equivalent to them?
1.Aisha takes the pond, n2 tiles, away from the large square of side n + 2. Expand: (n + 2)2 = n2 + 4n + 4, so (n + 2)2 − n2 = 4n + 4.
Aisha: the large square minus the pond. (n + 2)2 − n2 = n2 + 4n + 4 − n2 = 4n + 4. 2.Ben counts four runs of n + 1 tiles, each run being one side and one corner. Expand: 4(n + 1) = 4n + 4.
Ben: four runs of n + 1 tiles. 4(n + 1) = 4n + 4. 3.(a) Both expressions simplify to 4n + 4, so they are equal for every value of n. They are equivalent.
(a) Both expressions simplify to 4n + 4, so they are equivalent. 4.When n = 2, 4n + 4 = 12 and n2 + 8 = 4 + 8 = 12. One value that agrees does not prove that two expressions are equivalent, because two different expressions can be equal at a single value.
When n = 2 both 4n + 4 and n2 + 8 are 12. One value that agrees is not a proof. 5.Test another value. When n = 3, 4n + 4 = 16 but n2 + 8 = 9 + 8 = 17.
When n = 3, 4n + 4 = 16 but n2 + 8 = 17. 6.(b) Chen's expression is not equivalent to the others: when n = 3 it gives 17 tiles and the border needs 16. One value that disagrees is enough to show this.
(b) Chen's expression is not equivalent: when n = 3 it gives 17 tiles and the border needs 16.
Answer: (a) Both simplify to 4n + 4; (b) No: when n = 3 Chen's expression gives 17 tiles and the border needs 16
Watch out for
- Deciding that n2 + 8 is equivalent because it agrees when n = 2. Two different expressions can be equal at one value. Equivalent expressions are equal at every value.
- Expanding (n + 2)2 as n2 + 4. (n + 2)2 = (n + 2)(n + 2) = n2 + 2n + 2n + 4, so the 4n is part of it.
Your turn
1.Which is equivalent to ?