Introduction to Algebra

 

The objectives of this unit are to

*  solve problems involving number grids;

*  simplify simple algebraic expressions involving the addition, subtraction or multiplication of terms.

 

Number grids

 

 

 

6

9

12

8

11

14

10

13

16

 
 
The number grid below has two rules.  The across rule is to add 3.  The down rule is to add 2.

 

 

 


+ 2
 
6

 

+ 2
 
Right Arrow: We can complete the whole grid:

 

 

 

 

 

 

 

 

The next number grid has an across rule which is to subtract 3 and a down rule which is to add 4.

 

 

 


 + 4
 

 

 + 4
 
Right Arrow: We can complete the whole grid:

 

7

 

 

 

 

 

 

We can use a letter to represent the number in one grid in the number square.

For example, suppose a number grid has an across rule which is + 5 and a down rule which is to –2:

 

 

 


 - 2
 

 

 

 

 

 

 

 

 

+ 2
 

We could let the number in the top left-hand corner be n.  The whole grid would then be written as:

 

 


n

n + 5

n + 10

 - 2
 
n - 2

n + 3

n + 8

n - 4

n + 1

n + 6

 

 

Finding missing rules in number grids

 

Example:

Find the missing rules in and complete the following number grid.

 

 

 

 

 

 

?
 
19

 

 

7

 

 

 

 

 

 

17

 

 

Looking at the second row in the grid, we see that jumping 3 squares means subtracting 12.  So the across rule must be to subtract 4 each time.

 

So, the table now becomes:

- 4
 
 

 

 

 

 

 

?
 
19

15

11

7

 

 

 

 

 

 

17

 

 

Looking at the third column, we can see that jumping 2 squares down the table means adding 6.  So the down rule is to add 3 each time.

 

The complete table then is:

 

- 4
 
 

 

 


16

12

8

4

+3
 
19

15

11

7

22

18

14

10

25

21

17

13

 

 

 

Simplifying algebraic expressions

 

Examples involving addition and subtraction:

 

We can simplify expressions by collecting together terms that are alike:

 

n + n + n = 3n                                                  Add together all the n’s.

 

5n + 2 + 3n + 5 = 8n + 7                                  Add together the n’s and add the numbers.

 

6n + 7 – 5n + 2 = 1n + 9 = n + 9                      Note:  1n is written simply as n.

 

6e + 4 + 2e – 3 – 5e – 1 = 3e + 0 = 3e            We don’t need the + 0

 

4g + 7 + g – 4 – 8 = 5g – 5

 

6 + 4f – 2 – 9f + 1 = 5 – 5f

 

7 – 2k + 1 + 4k = 8 + 2k

 

4r – 5 + 2r – 3 = 6r – 8

 

6 – 4t – 2 – 5t = 4 – 9t

 

4r – 2 – 5r + 6 + r = 0r + 4 = 4                        We haven’t any r so we don’t need them in the answer.

 

Questions:

 

Simplify each of these expressions

 

6v + 2v + v =

 

 

2p + 3 + 5p – 7 =

 

5h + 4 + 2h + 6 =

 

 

4t – 2 + 6t + 8 =

 

b + 7 + 2b + 1 =

 

 

3a + 5 – 5a + 2 =

 

u + 6 + 3u + 2 + 4u =

 

 

4d + 2 – 5d + 1 =

 

3r + 5 + 2r – 6 =

 

 

6f – 3 + 2f + 8 =

 

5c + 6 – 4c + 2 =

 

 

5n – 4 + n – 5 =

 

2y + 7 + y + 1 + y =

 

 

6t – 2 + 7t – 9 =

 

3j + 5 – 2j + 1 – j =

 

 

8 – 4g + 2 + 7g =

 

6y + 2 – 5y – 2 =

 

 

1 – 12y + 4 + 8y =

 

3e + 5 + 6e – 6 =

 

 

6t – 4 + 2t + 4 – 8t =

 

 

Multiplying algebraic expressions

 

When multiplying, we multiply together any numbers present and we multiply together any letters. 

When we multiply something by itself we square it.

So n × n can be written as n²  (i.e. n squared).

 

Examples:

 

r × r = r²                                               Two r’s have been multiplied to it is r squared.

 

4 × t = 4t                                              Note: we don’t need to write the multiply sign in algebra.

 

7y × 2 = 14y                                        Multiply together the numbers.

 

5t × 6 × 2 = 60t

 

3 × 4k = 12k

 

6y × 2y = 18y²                                     Multiply together the numbers and letters.

 

5g × 3g = 15g²

 

6y × y = 6y²

 

2u × 8u × 3 = 48u²

 

 

Questions:

 

Simplify each of these expressions

 

t × t =

 

 

5r × 3r =

 

b × b =

 

 

4e × 4e =

 

5 × y =

 

 

3d × 2d =

 

3 × h =

 

 

6p × 7p =

 

4 ×2d =

 

 

5t × 7t =

 

6 × 6v =

 

 

8u × 2u × 5 =

 

7h × 4 =

 

 

4 × 2 × 3y =

 

2f × f =

 

 

6 × e × 9=

 

a × 5a =

 

 

4 × 5y × 4 =

 

j × 6j =

 

 

6 × 6t × 2t =

 

 

Mixed questions

 

Now try these questions – be careful not to mix up the adding/ subtracting methods with the multiplying method.

 

 

2w + 5 + 4w – 2  =

 

 

3p + 4 – 4p + 3 =

 

h × h =

 

 

6 – 2f + 1 + 5f =

 

4e + 2e + 3 + e  =

 

 

6y – 4 – 5y + 7 =

 

5t × 2 =

 

 

4t × 2t =

 

4r – 3 + r + 2 + 1 =

 

 

5 × 4f × 3 =

 

4f × 2f =

 

 

3u × 4 × 3u =

 

7j × 6 =

 

 

4c + 5 – 2c - 6 =

 

4d – 2 + 3d - 8 =

 

 

5 – 2h + 1 –h =

 

8h – 8 – 2h + 5 =

 

 

5 × 4q × 6q =

 

k × 4k × 3 =

 

 

6u + 7 – 5u + 1 – u =