Expressions and variables
What a letter like x stands for, how to put a number in its place, how to collect like terms, and how to turn words into algebra.
Algebra is arithmetic with letters. A letter like x stands for a number that we do not know yet, or for a number that can change. The word "algebra" comes from the title of a book written in Baghdad around the year 820 by the mathematician al-Khwarizmi. The habit of calling the unknown number x came much later, from the French thinker René Descartes in 1637. Letters let us write one short rule that works for every number, like "the price is 2 dollars a kilo plus 3 dollars for the bag". Below you learn four things: how to read algebra, how to put numbers in place of the letters, how to tidy up an expression, and how to turn words into algebra.
What a letter means
A letter that stands for a number is called a variable. A mix of numbers, letters and signs, like 2x + 3, is called an expression. The parts that are added or taken away are called terms. In 2x + 3 the terms are 2x and 3. The number in front of a letter is called the coefficient. In 2x the coefficient is 2.
Algebra has its own short way of writing. When a number and a letter stand side by side, they are multiplied. So 2x means "2 times x". When two letters stand side by side, they are multiplied too: ab means "a times b". A number next to a bracket is multiplied by what is inside: 3(4) means 3 times 4, which is 12. A fraction line means divide.
| Written | Read it as | If the letter is 4 |
|---|---|---|
| x + 5 | x plus 5 | 4 + 5 = 9 |
| 3x | 3 times x | 3(4) = 12 |
| x/2 | x divided by 2 | 4/2 = 2 |
| x² | x times x (the square of x) | 4(4) = 16 |
| 2x² | 2 times the square of x | 2(16) = 32 |
Putting a number in place of a letter
To find the value of an expression, you replace each letter with its number. This is called substitution. Write the number in brackets, so you do not glue it to the number in front. Then work it out in the usual order: squares first, then multiplying, then adding and taking away.
- Copy the expression, and put the number in brackets where each letter was.
- Work out any squares.
- Do the multiplying.
- Do the adding and taking away, from left to right.
Example 1
What is 3x − 5 when x = 6?
- Put 6 in place of x: 3(6) − 5.
- Multiply: 3(6) = 18.
- Take away: 18 − 5 = 13.
Answer: 13
Some expressions have two letters. You do exactly the same thing: each letter gets its own number.
Example 2
What is 3a − 2b when a = 5 and b = 4?
- Put the numbers in: 3(5) − 2(4).
- Multiply: 15 − 8.
- Take away: 15 − 8 = 7.
Answer: 7
The answer can be negative. If a = 2 and b = 7, then a − b = 2 − 7 = −5. You take away more than you have, so you go below zero.
Two letters side by side are multiplied. So ab with a = 4 and b = 6 is 4(6) = 24. It is not 46, and it is not 4 + 6 = 10.
Expressions with a square
The small 2 in x² means "multiply x by itself". It does not mean "times 2". Always work out the square before anything else.
Example 3
What is x² − 3x when x = 5?
- Put 5 in place of x: (5)² − 3(5).
- Square first: (5)² = 25.
- Multiply: 3(5) = 15.
- Take away: 25 − 15 = 10.
Answer: 10
Example 4
What is x² − 4 when x = −3?
- Put −3 in place of x, in brackets: (−3)² − 4.
- Square: (−3)² = (−3)(−3) = 9. A negative times a negative is positive.
- Take away: 9 − 4 = 5.
Answer: 5
Collecting like terms
Like terms have exactly the same letter part. 3x and 5x are like terms, because both are "some number of x". You can add them by adding the numbers in front: 3 of something plus 5 of the same thing is 8 of it. So 3x + 5x = 8x. The letter stays the same.
Remember
3x + 5x = 8x and 7a − 2a = 5a
Add or take away the coefficients. The letter part does not change.
Terms that are not alike cannot be joined. 3x + 5 stays as it is, because 5 is a plain number and not a number of x. And x and x² are not alike either: one is x, the other is x times x.
Example 5
Simplify 5x + 3 + 2x − 1.
- Find the like terms: 5x and 2x are the x terms; 3 and −1 are the plain numbers.
- Collect the x terms: 5x + 2x = 7x.
- Collect the numbers: 3 − 1 = 2.
- Write them together: 7x + 2.
Answer: 7x + 2
Turning words into algebra
Many questions are written in words. Each phrase has a short algebra version. Here are the ones you meet most.
| In words | In algebra |
|---|---|
| 5 more than x | x + 5 |
| 3 less than x | x − 3 |
| twice x | 2x |
| half of x | x/2 |
| the square of x | x² |
| x times y | xy |
| 4 less than twice x | 2x − 4 |
Try it yourself
1.What is 4x − 7 when x = 5?
Show the answer
13
4(5) = 20, and 20 − 7 = 13.
2.What is 2a + 3b when a = 4 and b = −1?
Show the answer
5
2(4) = 8 and 3(−1) = −3. So it is 8 − 3 = 5.
3.Simplify 6m + 4 − 2m + 3.
Show the answer
4m + 7
The m terms give 6m − 2m = 4m. The numbers give 4 + 3 = 7.
4.Which expression means "7 less than three times x"?
Show the answer
3x − 7
Three times x is 3x. Then take 7 away from it.
5.What is 3x² when x = 4?
Show the answer
48
Square first: 4² = 16. Then multiply by 3: 48. Not 12² = 144.
6.What is x² + 3x when x = −2?
Show the answer
−2
(−2)² = 4 and 3(−2) = −6. So it is 4 − 6 = −2.
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