Coefficients

In arithmetic 3 x 4 is a short way of writing 4 + 4 + 4. Similarly, in algebra, 3 x a is short for a + a + a. 3 x a is usually shortened to 3a.

Say this a ‘three a’.

3a = 3 x a = a + a + a

Notice the difference between 34 in arithmetic and 3a in algebra. 34 means 3 tens and 4 or 30 + 4. 34 does not mean 3 x 4. In algebra, 3a always means 3 x a.

3a, x + 1, 7 – r, 7y are all examples of algebraic expressions. In the expression 3a, the 3 is called the coefficient of a. The 3 shows that three a’s have been added together. For example

a. In 7y, 7 is the coefficient of y; seven y’s have been added together;

b. In x, there is only one x; x is the same as 1x; the coefficient of x is 1.

Note: It is usually to write x, not 1x.

Coefficients are not always whole number. Coefficients can also be fractions. In arithmetic 1/3 x 12 or 12/3 are short ways of writing 1/3 of 12 or 12 ÷ 3. In algebra 1/3a or a/3 are short ways of writing 1/3 of a or a ÷ 3.

In the expression 1/3a, the coefficient of a is the 1/3.

1/3a = a/3, so the coefficient of a in a/3 is also 1/3.

For example,

c. in 3/4x, ¾ is the coefficient of x;

d. in 2a/3, 2/3 is the coefficient of a ( since 2a/3 = 2/3a)

Grouping Positive and Negative Terms

Expressions such as 3x, x, 8x, 12x, 1/2x are called terms in x. We can add terms in x together.

3x + 2x = 5x

5x + 2x.

Thus two terms in x have been simplified to one term in x.

We can also subtract terms.

7y – 4y means 7 y’s take away 4 y’s.

This leaves 3 y’s. Thus

7y – 4y = 3y

Again, two terms have been simplified to one term.

We can simplify expressions which contain many terms. For example, the expression

3a = 8a + 5a + 9a – 2a

Means: add 3a, 5a, and 9a together; take away 8a and 2a.This gives 17a take away 10a. The result is 7a. We can write this as follows.

3a – 8a + 5a + 9a – 2a

= 3a + 5a + 9a – 8a – 2a

= 17a – 10a

= 7a

The first line in this example is important. We have grouped all the terms with a + sign before them together, and all the terms with a – sign before them together. This method is called grouping positive and negative terms.  It is usually the best way of simplifying the big expressions.

Simplify the following

a. 4a + 3a

b. 12d – 7d

c. 6m + 17m

Grouping Like and Unlike Terms

What is the sum of 5 plantains and 8 plantains?

13 plantains. Similarly, in algebra,

5p + 8p = 13p

What is the sum of 3 plantains and 7 torches?

All that can be said is that there is a mixture of plantain and torches. Similarly, in algebra it is impossible to simplify

3p + 7t

5p and 8p are like terms. Their sum is 13p. 3t and 7p are unlike terms. Their sum is 3p ­+ 7t…

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