ADDING AND SUBTRACTING ALGEBRAIC EXPRESSIONS  

Example 1 : 

Add 3p and 14p.

Solution :

= 3p + 4p

= 7p

Example 2 :

Add m, 12m and 21m.

Solution :

= m + 12m + 21m

= 34m

Example 3 :

Add 11abc and 5abc.

Solution :

= 11abc + 5abc

= 16abc

Example 4 :

Add 12y and -y.

Solution :

= 12y + (-y)

= 12y - y

= 11y

Example 5 :

Add 4x, 2x and -7x.

Solution :

= 4x + 2x + (-7x)

= 4x + 2x - 7x

= -x

Example 6 :

Add (pq - 1) and (3pq + 2).

Solution :

Step 1 : 

= (pq - 1) + (3pq + 2)

Step 2 :

Group the like terms 

= (pq + 3pq) + (-1 + 2)

Step 3 :

Simplify

= 4pq + 1

Example 7 :

Add (8x + 3) and (1 - 7x).

Solution :

Step 1 : 

= (8x + 3) + (1 - 7x)

Step 2 :

Group the like terms 

= (8x - 7x) + (3 + 1)

Step 3 :

Simplify

= x + 4

Example 8 :

Add (3x + ½) and (7x - 4½).

Solution :

Step 1 :

= (3x + ½) + (7x - 4½)

Step 2 :

Group the like terms.

= (3x + 7x) + (½ - 4½)

Step 3 :

Simplify.

= 10x - 4

Example 9 : 

Add (5x - 3y + 4z) and (1.5x + 0.4y + 8).

Solution :

Step 1 :

= (5x - 3y + 4z) + (1.5x + 0.4y + 8)

Step 2 :

Group the like terms 

= (5x + 1.5x) + (-3y + 0.4y) + 4z + 8

Step 3 :

Simplify

= 6.5x - 2.6y + 4z + 8

Example 10 : 

Add (3x - 4y + z) and (2x - z + 3y).

Solution :

Step 1 :

= (3x - 4y + z) + (2x - z + 3y)

Step 2 :

Group the like terms.

= (3x + 2x) + (-4y + 3y) + (z - z)

Step 3 :

Simplify.

= 5x + (-y) + 0

= 5x - y

Example 11 :

Subtract 7pq from 11pq.

Solution :

= 11pq - 7pq

= 4pq

Example 12 :

Subtract 3x²y from 5x²y.

Solution :

= 5x²y - 3x²y

= 2x²y

Example 13 :

Subtract -a from a.

Solution :

= a - (-a)

= a + a

= 2a

Example 14 : 

Subtract (5x + 7) from (21x + 9)

Solution :

Step 1 : 

= (21x + 9) - (5x + 7)

Distribute the negative sign to the terms in the second parenthesis.

= 21x + 9 - 5x - 7

Step 2 : 

Group the like terms. 

= (21x -5x) + (9 - 7)

Step 3 :

Simplify.

= 16x - 2

Example 15 : 

Subtract (1.5x + 1.4) from (-0.25x - 3).

Solution :

Step 1 : 

= (-0.25x - 3) - (1.5x + 1.4)

Distribute the negative sign to the terms in the second parenthesis.

= -0.25x - 3 - 1.5x - 1.4

Step 2 : 

Group the like terms.

= (-0.25x - 1.5x) + (-3 - 1.4)

Step 3 :

Simplify.

= -1.75x - 4.4

Example 16 : 

Subtract (2x - 5y) from (4x + 3y)

Solution :

Step 1 : 

= (4x + 3y) - (2x - 5y)

Distribute the negative sign to the terms in the second parenthesis.

= 4x + 3y - 2x + 5y

Step 2 : 

Group the like terms.

= (4x - 2x) + (3y + 5y)

Step 3 :

Simplify.

= 2x + 8y

Example 17 : 

Subtract (4b - 3a + c) from (2a - 3b + c). 

Solution :

Step 1 : 

= (2a - 3b + c) - (4b - 3a + c)

Distribute the negative sign to the terms in the second parenthesis.

= 2a - 3b + c - 4b + 3a - c

Step 2 : 

Group the like terms.

= (2a + 3a) + (-3b - 4b) + (c - c) 

Step 3 :

Simplify.

= 5a - 7b

Example 18 : 

Simplify :

100x + 99y – 98z + 10x + 10y + 10z – x – y + z

Solution :

Step 1 : 

= 100x + 99y – 98z + 10x + 10y + 10z – x – y + z

Group the like terms.

= (100x + 10x - x) + (99y + 10y - y) + (-98z + 10z + z) 

Step 3 :

Simplify.

= 109x + 109y + (-87z)

= 109x + 109y - 87z

Example 19 : 

John earns $20 on the first day and spends some amount in the evening. He earns $30 on the second day and spends double the amount as she spents on the first day. He earns $40 on the third day and spends 4 times the amount as she spents on the first day. Can you give an algebraic expression of the total amount with her, at the end of the third day?

Solution :

Step 1 : 

The amount earned on the first day is $20.

Let the amount spent on the first day be $x. Amount with him at the end of the first day is (20 - x).

Step 2 : 

Amount earned on the second day is $30 and the amount spent on the second day is $2x. The amount left on the second day is (30 - 2x).

Step 3 : 

The amount earned on the third day is $40.

Amount earned on the third day is $40 and the amount spent on the third day is $4x. The amount left on the third day is (40 - 4x).

Step 4 :

Therefore, the total amount that John would have at the end of three days is

= (20 - x) + (30 - 2x) + (40 - 4x)

= 20 - x + 30 - 2x + 40 - 4x

Group the like terms.

= (20 + 30 + 40) + (-x - 2x - 4x)

Simplify.

= 90 + (-7x)

= 90 - 7x

Example 20 : 

Jill and Kyle get paid per project. Jill is paid a project fee of $25 plus $10 per hour. Kyle is paid a project fee of $18 plus $14 per hour. Write an expression to represent how much a company will pay to hire both to work the same number of hours on a project.

Solution :

Step 1 : 

Write expressions for how much the company will pay each person. Let h represent the number of hours they will work on the project.

Jill : $25 + $10h     Kyle: $18 + $14h

  Fee + Hourly rate × Hours      Fee + Hourly rate × Hours

Step 2 : 

Add the expressions to represent the amount the company will pay to hire both.

Combine their pay : 

= 25 + 10h + 18 + 14h

Use the Commutative Property :

= 25 + 18 + 10h + 14h

Combine like terms :

= 43 + 24h

So, the company will pay 43 + 24h dollars to hire both Jill and Kyle.

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