**Add subtract and multiply linear expressions :**

Whenever we want add or subtract linear expressions, we must know about combining like terms.

**Like terms or Similar terms: **

Like terms are the terms which have the same variables with same exponent for each variable.

**Examples : **7x, 3x, - 4x

7x and -5xy are not like terms. Because in the first term we have the variable x only. But in the second term, we have x and y.So we cannot combine them.

To add or subtract linear expression, we have two methods

(i) Horizontal method

(ii) Vertical method

**Horizontal method : **

In this method, we arrange all the terms in a horizontal line and then add or subtract by combining the like terms.

**Vertical method :**

In this method, we should write the like terms vertically and then add or subtract.

Let us look into some example problems to understand how to add subtract and multiply linear expressions.

**Example 1 :**

Add 6a + 3 and 4a - 2.

**Solution :**

**Adding linear expressions in horizontal method :**

** = (6a + 3) + (4a - 2)**

** = 6a + 4a + 3 - 2**

** = 10a + 1**

**Adding linear expressions in vertical method :**

**Example 2 :**

Add 5y + 8 + 3z and 4y - 5

**Solution : **

**Adding linear expressions in horizontal method :**

** = (**5y + 8 + 3z) + (4y - 5)

** = **5y + 4y + 8 - 5 + 3z

** = 9**y + 3z + 3

**Adding linear expressions in vertical method :**

**Example 3 :**

Add 10x²- 5xy + 2y², -4x²+ 4xy + 5y², 3x²- 2xy - 6y²

**Solution : **

**Adding linear expressions in horizontal method :**

** = (**10x²- 5xy + 2y²) + (-4x²+ 4xy + 5y²) + (3x²- 2xy - 6y²)

** = **10x² - 4x² + 3x² - 5xy + 4xy - 2xy + 2y²+ 5y² - 6y²

= 13x² - 4x² - 7xy + 4xy + 7y² - 6y²

= 9x² - 3xy + y²

**Adding linear expressions in vertical method :**

**Example 4 :**

Subtract 6a - 3b from - 8a + 9b.

**Solution : **

**Subtracting linear expressions in horizontal method: **

** = (**- 8a + 9b) - (6a - 3b)

Distribute the negative sign inside the parentheses

** = **- 8a + 9b - 6a + 3b

= -8a - 6a + 9b + 3b

= -14a + 12b

**Subtracting linear expressions in vertical method :**

**Example 5 :**

Subtract a² + b² - 3ab from a² + b² - 3ab

**Solution : **

**Subtracting linear expressions in horizontal method: **

= (a² - b² - 3ab) - (a²+ b² - 3ab)

= a² - b² - 3ab - a² - b² + 3ab

= a² - a² - b² - b² - 3ab + 3ab

= - b² - b²

= - 2b²

**Subtracting linear expressions in vertical method :**

**Example 6 :**

Multiply (3a - 2b) (2p + 3q)

**Solution : **

By using distributive property we can multiply two polynomials.

= (3a - 2b) (2p + 3q)

= 3a(2p + 3q) - 2b(2p + 3q)

= 6ap + 9aq - 4pb - 6qb

Since there is not like terms, we cannot combine any more.

Hence the answer is 6ap + 9aq - 4pb - 6qb.

**Example 7 :**

Multiply (3x - 7) (7x - 3)

**Solution : **

By using distributive property we can multiply two polynomials.

= (3x - 7) (7x - 3)

= 3x(7x - 3) - 7(7x - 3)

= 21x² - 9x - 49x + 21

= 21x² - 58x + 21

Hence the answer is 21x² - 58x + 21.

After having gone through the stuff given above, we hope that the students would have understood "Add subtract and multiply linear expressions".

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