Equal expressions
Equal means the same value for every choice of numbers.
Two ways to write it
Rani writes the perimeter of a rectangle as l + b + l + b. Her friend writes 2l + 2b. They look different.
Are they equal?
What this lesson covers
The idea
Two algebraic expressions are equal when they take the same value for every choice of numbers for the letter-numbers; an expression obtained by applying the rules of numbers is equal to the original.
Two ways to write it
Rani writes the perimeter of a rectangle as l + b + l + b. Her friend writes 2l + 2b. They look different.
Are they equal?
- Yes: they have the same value whatever numbers l and b stand for
- No: they look different, so they are different
- Only for some numbers
Equal or not equal?
Put numbers in the letter-numbers and compare the values. Then decide: equal or not equal.
Same value for every choice
Two expressions are equal when they give the same value for every choice of numbers. One choice where they differ is enough to show they are not equal.
We can also be sure of equality when one expression comes from the other by the rules of numbers: swapping, grouping, the distributive property.
Two algebraic expressions are equal when they take the same value for every choice of numbers for the letter-numbers. An expression obtained from another by the rules of numbers is equal to the original.
- A | B
- l + b + l + b | 2l + 2b | equal
- 5u | 5 + u | not equal
- 10y − 3 | 10(y − 3) | not equal
Notes
Two algebraic expressions are equal when they take the same value for every choice of numbers. One that comes from another by the rules of numbers is equal to it.
Check yourself
x + 3 and 3 + x agree for *x* = 2, 5 and 9. Are they equal?
3z and z + 2 are both 3 when *z* = 1. Are they equal expressions?
2a + 2b is equal to a + b + a + b. Find the value of either when *a* = 5 and *b* = 1.
Answer: 12
2 × 5 + 2 × 1 = 10 + 2 = 12, and 5 + 1 + 5 + 1 = 12.
10y − 3 means “3 less than 10 times y”. What does 10(y − 3) mean?
- No. They look different.. Looking different is not enough: compare the values. Swapping terms never changes a value.
- Only when x is 3.. They agree for every number: the order of adding does not matter.
- Yes. Swapping the terms is a rule of numbers, so they agree for every x. — correct. Yes! The order of adding does not matter.
- Yes. They agree when z = 1.. Agreeing at one number is not enough.
- No. When z = 2 they are 6 and 4. — correct. Yes! Equal expressions must agree for every number. One difference is enough.
- Yes. One test is enough to be sure they are equal.. One test can show they are different, but it cannot show they are equal for all numbers.
- 3 less than 10 times y. That is 10y − 3.
- 10 times (3 less than y) — correct. Yes! First take 3 from y, then multiply by 10.
- 10 less than 3 times y. The 10 multiplies; it is not taken away.