Equivalent Expressions

1Equal for one value, or equivalent?

Two expressions are equivalent if they are equal for every value of the letter.

Look at x + 9 and 4x. When x = 3, both give 12. But when x = 1, x + 9 = 10 and 4x = 4. They are equal for one value only, so they are not equivalent.

2Ways to make equivalent expressions

1. Change the order of adding or multiplying. x + 3 is equivalent to 3 + x, and 4y is equivalent to y × 4.

2. Write repeated adding as multiplying. a + a + a + a + a = 5a.

3. Collect like terms. Like terms have exactly the same letter part. 3x + 4x = 7x, just as 3 books + 4 books = 7 books. Numbers go with numbers, x's with x's:
5a + 2b + 3a − b = (5a + 3a) + (2b − b) = 8a + b.
But 3x + 4 cannot be joined into 7x, because 3x and 4 are not like terms.

4. Use the distributive property. The number outside a bracket multiplies every term inside it:
3(x + 4) = 3 × x + 3 × 4 = 3x + 12.

You already use this in mental maths. Five bottles of juice at 490 Frw each:
5 × 490 = 5 × (500 − 10) = 2 500 − 50 = 2 450 Frw.

Picture a rectangle 3 units wide and (x + 4) units long. Its area can be found in one go, 3(x + 4), or in two parts, 3x + 12. Same rectangle, same area.

5. Factorise: the distributive property backwards. Find a number that divides every term and take it outside a bracket:
3n + 12 = 3(n + 4)
6a + 9 = 3(2a + 3)

3Checking by testing a number

Is 2(x + 3) equivalent to 2x + 3? Try x = 1: 2(1 + 3) = 8 but 2 × 1 + 3 = 5. They are different, so they are not equivalent. The correct expansion is 2(x + 3) = 2x + 6.

One number that gives different answers proves two expressions are not equivalent. To be sure they are equivalent, use the rules above (testing a few numbers is a good check, but it is not a proof).

Try it yourself

1. Simplify 4a + 6a.
2. Simplify 7x + 3 + 2x + 5.
3. Expand 5(y + 2).
4. Factorise 4m + 8.
5. Are 2(n + 5) and 2n + 5 equivalent?

Answers

1. 10a.
2. 9x + 8.
3. 5y + 10.
4. 4(m + 2).
5. No. Try n = 1: 2(1 + 5) = 12 but 2 × 1 + 5 = 7.