Exercise 10.4

1. Add the following algebraic expressions using both horizontal and vertical methods. Did you get the same answer with both methods?

(i) x2 - 2xy + 3y2; 5y2 + 3xy - 6x2

Horizontal Method:

We first arrange expressions in standard form.

x2−2xy+3y2 = x −xy+y2

5y2+3xy−6x2=−x2+xy+y2

Now, we add x2−2xy+3y2 and −6x2+3xy+5y2

x2−2xy+3y2+−6x2+3xy+5y2

Now, we add like terms.

x2−6x2+−2xy+3xy+3y2+5y2

= -x2+xy+y2

Vertical Method:

We first arrange expressions in standard form.

x2−2xy+3y2 = x2−2xy+3y2

5y2+3xy−6x2 = -x2+xy+y2

Now, we add x2−2xy+3y2 and −6x2+3xy+5y2

x2−2xy+3y2 + (−6x2+3xy+5y2)

-x2+xy+y2

By both methods we get the same answer.

(ii) 4a^2 + 5b^2 + 6ab ; 3ab ; 6a^2 – 2b^2 ; 4b^2 – 5ab

Horizontal Method:

We first arrange expressions in standard form.

4a^2 + 5b^2 + 6ab = input(4)a^2 + input(5)b^2 + input(6)ab

3ab = input(3)ab

6a^2 – 2b^2 = input(6)a^2 - input(2)b^2

4b^2 – 5ab = input(4)b^2 - input(5)ab

Now, we add all the expressions:

4a^2 + 5b^2 + 6ab + 3ab + 6a^2 - 2b^2 + 4b^2 - 5ab

Now, we add like terms.

(4a^2 + 6a^2) + (5b^2 - 2b^2 + 4b^2) + (6ab + 3ab - 5ab)

= input(10)a^2 + input(7)b^2 + input(4)ab

Vertical Method:

We first arrange expressions in standard form.

4a^2 + 5b^2 + 6ab = 4a^2 + 5b^2 + 6ab

3ab = 3ab

6a^2 – 2b^2 = 6a^2 - 2b^2

4b^2 – 5ab = 4b^2 - 5ab

Now, we add all the expressions:

4a^2 + 5b^2 + 6ab
+ 3ab

• 6a^2 - 2b^2
  + 4b^2 - 5ab

10a^2 + 7b^2 + 4ab

By both methods we get the same answer.

(iii) 2x + 9y – 7z ; 3y + z + 3x ; 2x – 4y – z

Horizontal Method:

We first arrange expressions in standard form.

2x + 9y – 7z = input(2)x + input(9)y - input(7)z

3y + z + 3x = input(3)x + input(3)y + input(1)z

2x – 4y – z = input(2)x - input(4)y - input(1)z

Now, we add all the expressions:

2x + 9y - 7z + 3x + 3y + z + 2x - 4y - z

Now, we add like terms.

(2x + 3x + 2x) + (9y + 3y - 4y) + (-7z + z - z)

= input(7)x + input(8)y - input(7)z

Vertical Method:

We first arrange expressions in standard form.

2x + 9y – 7z = 2x + 9y - 7z

3y + z + 3x = 3x + 3y + z

2x – 4y – z = 2x - 4y - z

Now, we add all the expressions:

2x + 9y - 7z
3x + 3y + z
2x - 4y - z

7x + 8y - 7z

By both methods we get the same answer.

(iv) 2x^2 – 6x + 3 ; –3x^2 – x – 4 ; 1 + 2x – 3x^2

Horizontal Method:

We first arrange expressions in standard form.

2x^2 – 6x + 3 = input(2)x^2 - input(6)x + input(3)

–3x^2 – x – 4 = -input(3)x^2 - input(1)x - input(4)

1 + 2x – 3x^2 = -input(3)x^2 + input(2)x + input(1)

Now, we add all the expressions:

2x^2 - 6x + 3 - 3x^2 - x - 4 - 3x^2 + 2x + 1

Now, we add like terms.

(2x^2 - 3x^2 - 3x^2) + (-6x - x + 2x) + (3 - 4 + 1)

= -input(4)x^2 - input(5)x + input(0)

Vertical Method:

We first arrange expressions in standard form.

2x^2 – 6x + 3 = 2x^2 - 6x + 3

–3x^2 – x – 4 = -3x^2 - x - 4

1 + 2x – 3x^2 = -3x^2 + 2x + 1

Now, we add all the expressions:

2x^2 - 6x + 3
-3x^2 - x - 4
-3x^2 + 2x + 1

-4x^2 - 5x + 0

By both methods we get the same answer.