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Lesson 22 · Algebraic Expressions

Introducing Algebraic Expressions

MathematicsSubject
14 minEstimated read
A letter is a number you do not know yet

A letter holds a place for a number, so you can work with the number before you know it.

The words you need

Word What it means In 5x² − 3x + 7
TermA part separated by a plus or minus signThree terms
CoefficientThe number in front of the letter5 and −3
VariableThe letter itselfx
ConstantA term with no letter in it7
Like terms are terms of the same kind

Like terms have exactly the same letters raised to exactly the same powers. 3x and 3x² are not like terms.

Only like terms can be added

3x plus 5x is 8x, just as 3 apples plus 5 apples is 8 apples. But 3x plus 5y stays as it is.

3x plus 5y is not 8xy

If x is 2 and y is 4, then 3x + 5y is 26 while 8xy would be 64. Substituting numbers settles any doubt.

Substitution

Replace each letter by its number, doing the multiplications before the additions.

2x does not mean 2 next to x

2x means 2 multiplied by x, so when x is 3 the value is 6 and not 23. The multiplication sign is always there, just not written.

Turning words into expressions

Five less than a number is x minus 5, not 5 minus x. Naming the letter first prevents most of the errors.

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Worked Example 1

Simplify 7x + 3y − 2x + 5y − 4.

Sort the terms into kindsx terms: 7x and −2x. y terms: 3y and 5y. Constant: −4
Combine each kind5x, 8y and −4
Write the answer5x + 8y − 4
Check with numbersWith x = 1 and y = 1, both give 9. That check takes ten seconds.
Worked Example 2

If a = 4 and b = −3, find the value of 3a² − 2ab.

Replace the letters, using brackets3 × 4² − 2 × 4 × (−3)
Powers before multiplication4² = 16, so the first term is 48
The second term2 × 4 × (−3) = −24, and subtracting −24 adds 24
Answer72
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Activity 22.1

The matchstick pattern

What you needAbout thirty matchsticks, or thin strips of card cut to the same length.
MethodMake one square, then add squares in a row, counting the matchsticks each time.
Then predictWork out how many a row of ten would need, then write an expression for n squares.
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1

A letter stands for a number you do not know yet, so you can work with it early.

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1What is a letter doing in an algebraic expression?
2Why can 3x + 5y not be simplified, when 3x + 5x can?
3Why are 3x and 3x² not like terms?
4How do you substitute numbers into an expression safely?
5How do you turn a sentence in words into an algebraic expression?

Question 1 of 10

1In algebra, a letter stands for
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Slide 1 of 4
Unit 22 · Algebraic Expressions

Algebraic Expressions

Mathematics · Grade 7

What You Will Be Able To Do

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Read an expression: terms, coefficients, variables and constants

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Collect like terms, and know why unlike terms will not combine

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Substitute numbers, including negative ones, without losing a sign

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Turn a sentence in words into an expression that says the same thing

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A letter holds a place for a number, so you can work with the number before you know it.

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Presenter notes: Open with a purchase the class cannot complete. Say you bought three pens but you have not looked at the price yet, and ask how much you spent. Somebody will say three times the price, which is exactly right and is already algebra. Write it as 3x on the board and point out that they have just done the thing the whole unit is about: working with a number before knowing it.